! 


i 


IN   MEMORIAM 
FLORIAN  CAJORl 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/advancedarithmetOOIymarich 


Advanced  arithmetic 


BY 


ELMER   A.    LYMAN 

PROFESSOR   OF  MATHEMATICS   IN   THE   MICHIGAN   STATE 
NORMAL   COLLEGE,    YPSILANTI,    MICHIGAN 


5»<C 


NEW  YORK  •:.  CINCINNATI  •:.  CHICAGO 

AMERICAN    BOOK     COMPANY 


Copyright,  1905,  by 
ELMEE   A.   LYMAN. 

Entered  at  Stationers'  Hall,  London. 


LYMAN,    ADV.    ARITII. 
W.   P.     I 


r\ 


Qr\,U'l 


u^ 


PREFACE 

The  need  felt  for  an  advanced  text-book  in  arithmetic 
that  shall  develop  fundamental  princi2)les  and  at  the  same 
time  include  the  essentials  of  commercial  practice  is  re- 
sponsible for  the  appearance  of  this  book.  Tlie  author 
believes  that  mental  training  is  an  important  feature  in 
the  study  of  arithmetic,  but  that  the  study  need  lose  none 
of  this  training  by  the  introduction  of  practical  business 
methods.  Consequently,  throughout  the  work,  the  aim 
has  been  not  only  to  develop  the  principles  of  the  subject, 
both  by  means  of  demonstrations  and  exercises,  but  also 
to  employ  such  methods  and  short  processes  as  are  used 
in  the  best  commercial  practice,  and  to  exclude  cumber- 
some methods  and  useless  material. 

The  book  is  intended  for  pupils  who  have  completed 
the  grammar  school  work  in  arithmetic,  and  contains 
abundant  material  for  a  review  and  advanced  course. 

The  exercises  have  been  selected  largely  from  actual 
business  transactions.  A  few  have  been  taken  from 
standard  foreign  works. 

Brief  historical  notes  are  occasionally  inserted  in  the 
hope  that  they  will  be  of  interest  and  value. 

The  author  is  indebted  to  several  friends,  who,  after 
careful  reading  of  manuscript,  or  proof  sheets,  or  both, 
have  offered  valuable  suggestions. 

E.  A.  LYMAN, 


=-=«>r>ertf^o«2 


CONTENTS 

PAGE 

Notation  and  Numeration 7 

Addition       .         .         .         .        / 12 

Subtraction 17 

Multiplication 22 

Division 28 

Factors  and  Multiples    «         .......  32 

Casting  out  Nines     .         .        o 40 

Fractions 46 

Approximate  Results 57 

Measures 62 

Longitude  and  Time 78 

The  Equation 86 

Powers  and  Roots 89 

Mensuration 99 

Graphical  Representations    .        .        .    '    .        .        .        .  124 

Ratio  and  Proportion lol 

Method  of  Attack 140 

Percentage .    .    .152 

Commercial  Discounts 157 

Marking  Goods  .        .        . 162 

Commission  and  Brokerage 164 

6 


6  CONTENTS 

pAot": 

Interest „         .        .         .         .  167 

Banks  and  Banking .         .         .  186 

Exchange ,        .        .        .  193 

Stocks  and  Bonds ,        .        .  200 

Insurance 207 

Taxes  and  Duties 216 

The  Progressions .  219 

Logarithms 224 

Exercises  for  Review 235 


ADYAKCED    ARITHMETIC 


NOTATION   AND   NUMERATION 

1.  Our  remote  ancestors  doubtless  did  their  counting 
by  the  aid  of  the  ten  fingers.  Hence,  in  numeration  it 
became  natural  to  divide  numbers  into  groups  of  tens. 
This  accounts  for  the  almost  universal  adoption  of  the 
decimal  scale  of  notation. 

2.  It  is  uncertain  what  the  first  number  symbols  were. 
They  were,  probably,  fingers  held  up,  groups  of  pebbles, 
notches  on  a  stick,  etc.  Quite  early,  however,  groups  of 
strokes  I,  II,  III,  llll,  •••,  were  used  to  represent  numbers. 

3.  The  earliest  written  symbols  of  the  Babylonians 
were  cuneiform  or  wedge-shaped  symbols.  The  vertical 
wedge  (I)  was  used  to  represent  unity,  the  horizontal 
wedge  (— ')  to  represent  ten,  and  the  two  together  (f*— ) 
to  represent  one  hundred.  Other  numbers  were  formed 
from  these  symbols  by  writing  them  adjacent  to  each  other. 
Thus, 

yrr  =1  +  1  +  1  =  3, 
-^-^rr  =  10  + 10  + 1  + 1  =  22, 

^]^  =10x100  =  1000, 

\\1^ ^)  =  5  X  100  +  10  +  2  =  512. 

7 


8  NOTATION  AND  NUMERATION 

To  form  numbers  less  than  100  the  symbols  were  placed 
adjacent  to  each  other  and  the  numbers  they  represented 
were  added.  To  form  numbers  greater  than  100  the  sym- 
bols representing  the  number  of  hundreds  were  placed 
at  the  left  of  the  symbol  for  one  hundred  and  used  as  a 
multiplier. 

4.  The  Egyptians  used  hieroglyphics^  pictures  of  objects, 
or  animals  that  in  some  way  suggested  the  idea  of  the 
number  they  wished  to  represent.  Thus,  one  was  repre- 
sented by  a  vertical  staff  (I),  ten  by  a  symbol  shaped  like 
a  horseshoe  (o),  one  hundred  by  a  short  spiral  (^),  one 
hundred  thousand  by  the  picture  of  a  frog,  and  one  million 
by  the  picture  of  a  man  with  outstretched  hands  in  the 
attitude  of  astonishment.  They  placed  the  symbols  adja- 
cent to  each  other  and  added  tlieir  values  to  form  other 
numbers.  Thus,  ^oi  =  100  -f  10  +  1  =  HI.  The  Egyp- 
tians had  other  symbols  also. 

5.  The  Greeks  used  the  letters  of  their  alphabet  for 
number  symbols,  and  to  form  other  numbers  combined 
their  symbols  much  as  the  Babylonians  did  their  wedge- 
shaped  symbols. 

6.  The  Romans  used  letters  for  number  symbols,  as 
follows : 


1 

5 

10 

50 

100 

500 

1000 

I 

V 

X 

L 

c 

D 

M 

Numbers  are  represented  by  combinations  of  these  sym- 
bols according  to  the  following  principles : 

(1)  The  repetition  of  a  symbol  repeats  the  value  of  the 
number  represented  by  that  symbol;  as,  111  =  8,  XX  =  20. 

(2)  The  value  of  a  niuuber  is  diminished  by  placing 
a  symbol  of  less  value  before  one  of  greater  value  ;    as, 


NOTATION  AND  NUMK RATION  9 

IV  =  4,   XL  =  40,   XC  =  90.      The   less    lunubei'    is   sub- 
tracted from  the  greater. 

(3)  The  value  of  a  number  is  increased  by  placing  a 
symbol  of  less  value  after  one  of  greater  value,  as  XI  =  11, 
CX  =  110.  The  less  number  is  added  to  the  greater 
number. 

(4)  The  value  of  a  number  is  midtipUed  by  1000  by 
placing  a  bar  over  it,  as  C  =  100,000,  X  =  10,000. 

7.  Among  the  ancients  we  do  not  find  the  character- 
istic features  of  the  Arabic,  or  Hindu  system  where  each 
symbol  has  two  values,  its  intrinsic  value  and  its  local  value^ 
i.e.  the  value  due  to  the  position  it  occupies.  Thus,  in  the 
number  513  the  intrinsic  value  of  the  symbol  5  is  five,  its 
local  value  is  five  hundred.  Written  in  Roman  notation 
513  =  DXIII.  In  the  Roman  notation  each  symbol  has 
its  intrinsic  value  only. 

8.  The  ancients  lacked  also  the  symbol  for  zero,  or  the 
absence  of  quantity.  The  introduction  of  this  symbol 
made  place  value  possible. 

9.  With  such  cumbersome  symbols  of  notation  the  an- 
cients found  arithmetical  computation  very  difficult.  In- 
deed, their  symbols  were  of  little  use  except  to  record 
numbers.  The  Roman  symbols  are  still  used  to  number 
the  chapters  of  books,  on  clock  faces,  etc. 

10.  The  Arabs  brought  the  present  system,  including 
the  symbol  for  zero  and  place  value,  to  Europe  soon  after 
the  conquest  of  Spain.  This  is  the  reason  that  the  nu- 
merals used  to-day  are  called  the  Arabic  numerals.  The 
Arabs,  however,  did  not  invent  the  system.  They  received 
it  and  its  figures  from  the  Hindus. 


10 


NOTATION  AND  NUMERATION 


11.  The  origin  of  each  of  the  number  symbols  4,  5,  6,  7, 
9,  and  probably  8  is,  according  to  Ball,  the  initial  letter  of 
the  corresponding  numeral  word  in  the  Indo-Bactrian  al- 
]3habet  in  use  in  the  north  of  India  about  150  B.C.  2  and  3 
were  formed  by  two  and  three  parallel  strokes  written 
cursively,  and  1  by  a  single  stroke.  Just  when  the  zero 
was  introduced  is  uncertain,  but  it  probably  appeared 
about  the  close  of  the  fifth  century  a.d.  The  Arabs 
called  the  sign  0,  sifr  (sifra  =  empty).  This  became  the 
English  cipher  (Cajori,  ''History  of  Elementary  Mathe- 
matics"). 

12.  The  Hindu  system  of  notation  is  capable  of  unlimited 
extension,  but  it  is  rarely  necessary  to  use  numbers  greater 
than  billions. 

13.  In  the  development  of  any  series  of  number  sym- 
bols into  a  complete  system,  it  is  necessary  to  select  some 
number  to  serve  as  a  base.  In  the  Arabic,  or  Hindu  system 
ten  is  used  as  a  base;  i.e.  numbers  are  written  up  to  10, 
tlien  to  20,  then  to  30,  and  so  on.  In  this  system  9  digits 
and  0  are  necessary.  If  five  is  selected  as  the  base,  but 
4  digits  and  0  are  necessary.  If  twelve  is  selected,  11 
digits  and  0  are  necessary. 

The  following  table  shows  the  relations  of  numbers  in 
the  scales  of  10,  5,  and  12.  (t  and  e  are  taken  to  represen.t 
ten  and  eleven  in  the  scale  of  12.) 


Bask 

12 

21 

48 

10 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

100 

5 

1 

1 

2 
2 

3 
3 

4 
4 

10 
5 

11 
6 

12 

7 

13 

8 

14 
9 

20 

t 

21 

e 

22 
10 

41 
19 

143 

400 

12 

40 

84 

NOTATION  AND  NUMERATION  11 


Ux.  1.    Reduce  431^  to  tlie  decinjal  scale. 

Note.     431,  means  431  in  the  scale  of  5. 

Solution.     4  represents  4  x  5  x  5  =  100 
3  represents  3x5  =15 

1  represents  1  =1 


..  431, 

=  116,, 

Fx.  2. 

Reduce  46^2^^ 

to  the 

scale  of  8. 

Solution. 

4632 

579  0  =  579  units  of  the  second  order  and  none  of  the  first  order. 

72  3  =    72  nnits  of  the  third  order  and  3  of  the  second  order. 

9  0  =      9  units  of  the  fourth  order  and  none  of  the  third  order. 

1  1  =      1  unit    of  the  fifth  order  and  1  of  tlie  fourth  order. 

.-.  4632io  =  llOSOg. 

EXERCISE   1 

1.  What  number  symbols  are  needed  for  the  scale  of  2? 
of  8?  of  6  ?  of  11  ?     Write  12  and  20  in  the  scale  of  2. 

2.  Reduce  2345  and  54(3^  to  the  decimal  scale. 

3.  Reduce  7649^^  to  the  scale  of  4. 

4.  Compare  the  local  values  of  the  two  9's  in  78,940,- 
590,634.  What  is  the  use  of  the  zero  ?  Why  is  the  num- 
ber grouped  into  periods  of  three  figures  each  ?     Read  it. 

5.  If  4  is  annexed  to  the  right  of  376,  how  is  the  value 
of  each  of  the  digits  3,  7,  6  affected  ?  if  4  is  annexed  to 
the  left  ?  if  4  is  inserted  between  3  and  7  ? 

6.  What  is  the  local  value  of  each  figure  in  76,345  ? 
What  would  be  the  local  value  of  the  next  figure  to  tlie 
right  of  5  ?  of  the  next  figure  to  the  right  of  this? 

7o    For  what  purpose  is  the  decimal  point  used  ? 

8.    Read  100.004  and  0.104  ;  0.0002  ;  0.0125  and  100.0025. 


ADDITION 

14.  If  the  arrangement  is  left  to  the  computer,  numbers 
to  be  added  shoukl  be  written  in  columns  with  units  of 
like  order  under  one  another. 

15.  In  adding  a  column  of  given  numbers,  the  computer 
should  think  of  results  and  not  of  the  numbers. 

He  should  not  say  three  and  two  are  five  and  one  are  six  329 

and  four  are  ten  and  nine  are  nineteen,  but  sinij)ly  five,  six,  764 

ten,  nineteen,  writing  down  the  9  as  he  names  tlie  last  nuni-  221 

ber.     The  remaining  columns  should  be  added  as  follows:  9642 

three,   seven,  nine,  fifteen,  seventeen,  writing   down   the  7 ;  7823 

nine,  fifteen,    seventeen,   twenty-four,   twenty-seven,  writing  18779 

down  tlie  7  ;  nine,  ei/z/i^e^??,  writing  down  the  18.     Time  in  211 
looking  for  errors  may  be  saved  by  writing  the  numbers  to 
be  carried  underneath  the  sum  as  in  the  exercise. 

16.  Checks.  If  the  columns  of  figures  have  been  added 
upward,  check  by  adding  downward.  If  the  two  results 
agree,  the  work  is  probably  correct. 

Another  good  check  for  adding,  often  used  by  account- 
ants, is  to  add  beginning  with  the  left-hand  column. 


Thus,  the  sum  of  the  thousands  is  16  thou- 
sands, of  the  hundreds  26  hundreds,  of  the  tens 
16  tens,  and  of  the  units  19  units. 

EXERCISE   2 


16000 

or 

16 

2600 

26 

160 

16 

19 

19 

18779  18779 


1.    What  is   meant  by   the   order   of  a  digit?     Define 
addend^  sum. 

12 


ADDITION  13 

2.  Why  should  digits  of  like  order  be  placed  in  tlie 
same  column  ?     State  the  general  principle  involved. 

3.  Wliy  should  the  columns  be  added  from  right  to 
left  ?  Could  the  columns  be  added  from  left  to  right  and 
a  correct  result  be  secured?  What  is  the  advantage  in 
beginning  at  the  right  ? 

4.  In  the  above  exercise,  why  is  1  added  (''carried") 
to  the  second  column?  1  to  the  third  column?  2  to  the 
fourth  column  ? 

17.  Accuracy  and  rapidity  in  computing  should  be  re- 
quired from  the  first.  Accuracy  can  be  attained  by  acquir- 
ing the  habit  of  ahvays  checking  results.  Rapidity  comes 
with  much  practice. 

18.  The  45  simple  combinations  formed  by  adding  con- 
secutively each  of  the  numbers  less  than  10  to  itself  and 
to  every  other  number  less  than  10  should  be  practiced  till 
the  student  can  announce  the  sum  at  sight.  These  com- 
binations should  be  arranged  for  practice  in  irregular  order 
similar  to  the  following : 

122598145764234 
123138976869464 


6 

6 

1 

1 

3 

4 

9 

5 

2 

7 

1 

2 

5 

3 

5 

9 

7 

2 

8 

7 

5 

9 

8 

7 

9. 

6 

8 

r 
O 

4 

7 

5 

4 

2 

2 

6 

4 

3 

1 

2 

1 

8 

3 

1 

3 

7 

9 

8 

5 

9 

8 

6 

5 

3 

6 

7 

9 

3 

4 

8 

7 

19.    Rapid  counting  by  ones,  twos,  threes,  etc.,  up  to  nines 
is  very  helpful  in  securing  both  accuracy  and  rapidity. 

Ex.    Begin  with  4  and  add  6's  till  the  result  equals  100. 
Add  rapidly,  and  say  simply  4,  10,  16,  22,   •   •   •,  94,  100. 


14  ADDITION 

20.    It  is  helpful  also  to  know  combinations,  or  groups 

12      3      4      5, 

that  form  certain  numbers,      inus,   n      o      rr      ^      r   ^nd 

y      8      7      b      o 

8  7     6     6     5     5     4 

12     2     3     4     3     3,  etc.,  are  groups  that  form  10,  and 
112     112     3 

99998887 

9  8     7     6     8     7     6     7    are  groups  that  form  20. 
23454566 


21.  Such  groups  should  be  carefully  studied  and  prac- 
ticed until  the  student  readily  recognizes  them  in  his 
work.  He  should  also  familiarize  himself  with  other 
groups..  The  nine-groups  and  the  eleven-groups  are  easy 
to  add,  since  adding  nine  to  any  number  diminishes  the 
units'  figure  by  one,  and  adding  eleven  increases  the  units' 
and  the  tens'  digits  each  by  one. 

EXERCISE   3 

1.  Begin  with  8  and  add  7's  till  the  result  is  50. 

2.  Begin  with  3  and  add  8's  till  the  result  is  67. 

Form  the  following  sums  till  the  result  exceeds  100  ,- 

3.  Begin  with  3  and  add  7's. 

4.  Begin  with  7  and  add  8's. 

5.  Begin  with  5  and  add  9's. 

6.  Begin  witli  8  and  add  5's. 

7.  Beofin  with  5  and  add  6's. 

8.  Begin  with  6  and  add  3's. 


A  DDITJON 


15 


Add  the  following'  coliniiiis,  beginning'  at  tlie  bottom, 
and  check  the  results  by  adding  downward.  Form  such 
groups  as  are  convenient  and  add  them  as  a  single  number. 
In  the  first  two  exercises  groups  are  indicated. 


9.     10. 

11. 

12. 

13. 

14. 

15. 

16. 

re 

7 

5 

25.4 

2122 

275 

5427 

47.683 

3   1 

r9 

[1 

4 

76.1 

7642 

267 

6742 

72.125 

1 

1 

34.59 

8321 

979 

8374 

94.467 

'8 

r4 

6 

43.33 

9789 

231 

9763 

53.2124 

5 

8 

67.27 

2432 

486 

2134 

91.576 

.2 

■'- 

4 

81.2 

5765 

523 

bm'o 

14.421 

'9 

4 

2 

28.3 

1297 

752 

3249 

32.144 

^'    i 

I  2 

9 

32.99 

6423 

648 

1678 

67.6797 

o        1 

7 

16.25 

1678 

486 

2432 

19.045 

8 

8 

4 

53.11 

3212 

529 

5469 

54.091 

'9 

8 

7 

3 

91.5 

7679 

926 

8761 

86.2459 

4 

2 

85.4 

2144 

842 

2332 

27.654 

2 

4 

1 

74.1 

1576 

236 

5467 

98.346 

1 

5 

5 

22.22 

4467 

574 

1023 

84.6211 

In  commercial  operations  it  is  sometimes  convenient  to 
add  numbers  written  in  a  line  across  the  page.  If  totals 
are  required  at  the  right-hand  side  of  tlie  page,  add  from 
left  to  right  and  check  by  adding  from  right  to  left. 

Add  : 

17.  23,  42,  31,  76,  94,  11,  13,  27,  83,  62,  93. 

18.  728,  936,  342,  529,  638,  577,  123,  328,  654. 

19.  1421,  2752,  7846,  5526,  3425,  1166,  7531,  8642. 

20.  46,  72,  88,  44,  39,  37,  93,  46,  64,  73,  47. 

21.  1728,  3567,  2468,  5432,  4567,  2143,  9876,  6789. 


16  ADDITION 

Find  the  sum  of  the  following  numbers  by  adding  the 
columns  and  then  adding  the  results  horizontally.  Check 
by  adding  the  rows  horizontally  and  then  adding  the 
columns  of  results. 


22.  7642 

6241 

5331 

3124 

4724 

8246 

9372 

3623 

2793 

51096 

23.  793 

864 

927 

531 

642 

876 

927 

426 

459 

24.  7942 

8349 

2275 

3673 

9527 

2136 

3411 

4212 

6524 

7641 

5675 

7987 

3171 

1234 

2892 

6425 

25.  26 

72 

126 

467 

354 

987 

54 

86 

13 

34 

45 

56 

67 

67 

87 

43 

98 

87 

765 

453 

342 

465 

783 

5 

21 

5 

43 

350 

9 

11 

321 

24 

8 

25 

196 

961 

649 

378 

452 

36 

77 

66 

555 

444 

888 

999 

111 

222 

Exercises  for  further  practice  in  addition  can  be  readily  supplied 
by  the  teacher.  The  student  should  be  drilled  till  he  can  add  accu- 
rately and  rapidly.  Accuracy,  however,  should  never  be  sacrificed  to 
attain  rapidity. 

Expert  accountants,  by  systems  of  grouping  and  much  pi-actice, 
acquire  facility  in  adding  two  or  even  three  colunms  of  figures  at  a 
time.  Elaborate  calculating  machines  have  also  been  invented,  and 
are  much  used  in  banks  and  counting  offices.  By  means  of  these 
machines,  columns  of  numbers  can  be  tabulated  and  the  sum  printed 
by  simply  turning  a  lever. 


SUBTRACTION 

22.  In  subtraction  it  is  important  that  the  student  shoukl 
be  able  to  see  at  once  what  number  added  to  the  smaller 
of  two  numbers  of  one  figure  each  will  produce  the  larger. 
Thus,  if  the  difference  between  5  and  9  is  desired,  the 
student  should  at  once  think  of  4,  the  number  which 
added  to  5  produces  9. 

23.  Again,  if  the  second  number  is  the  smaller,  as  in 
7  from  5,  the  student  should  think  of  8,  the  number 
which  added  to  7  produces  15,  the  next  number  greater 
than  7  which  ends  in  5. 

24.  The  complete  process  of  subtraction  is  shown  in  the 
following  exercise  : 

8534      "^  ^^^^^  -  ^^^  ^^^  carry  1.     (Why  carry  1  ?) 
.go^      3  and  0  are  3. 

■ 6  and  9  are  15,  carry  1. 

2907      p        1  o         Q 
b  and  2  are  8. 

25.  The  student  should  think  "  What  number  added  to 
5627  will  produce  8534  ?  "  After  a  little  practice,  it  is 
unnecessary  to  say  more  than  7  and  7,  3  and  0,  6  and  9, 
6  and  2,  writing  down  the  underscored  digit  just  as  it  is 
named. 

26.  Check.  To  check,  add  the  remainder  and  the  sub- 
trahend upward,  since  in  working  the  exercise  the  numbers 
were  added  downward. 

LTMAX'S  ADV.    AR.  —2  17 


18  SUBTRACTION 

27.  The  above  method  of  subtraction  is  important  not 
only  because  it  can  be  performed  rapidly,  but  because  it 
is  very  useful  in  long  division.  It  is  also  the  method  of 
"  making  change  "  used  in  stores. 

28.  There  are  two  other  metliods  of  subtraction  in 
common  use.     The  processes  are  shown  in  the  following 

exercises : 

(1)  643  =  600  +  40  +  3  zr  500  +  130  +  13 
456  =  400  +  50  +  6  =  400  +  50+6 
187=      ~  100+    80+7 

6  from  13,  7 ;  5  from  13,  8 ;  4  from  5,  1. 

(2)  643  =  600  +  40  +  3,  600  +  140  +  13 
456  =  400  +  50  +  6,  500+  60+6 
187=  100+    80+7 

6  from  13,  7 ;  6  from  14,  8 ;  5  from  6,  1. 

EXERCISE  4 

1.  Define  the  terms  subtrahend^  minuend^  difference. 

2.  How  should  the  terms  be  arranged  in  subtraction  ? 
Where  do  we  begin  to  subtract  ?     Why  ? 

3.  Is  the  difference  affected  by  adding  the  same  number 
to  both  subtrahend  and  minuend  ?  Is  this  principle  used 
in  either  (1)  or  (2)  ? 

4.  If  a  digit  in  the  minuend  is  less  than  a  digit  of  the 
corresponding  order  in  the  subtrahend,  explain  how  the 
subtraction  is  performed  in  both  (1)  and  (2). 

29.  Arithmetical  Complement.  The  arithmetical  com- 
plement of  a  number  is  the  difference  between  the  number 
and  the  next  higher  power  of  10.  Thus,  the  arithmetical 
complement  of  642  is  358,  since  358  +  642  =  1000.  The 
arithmetical  complement  of  0.34  is  0.66,  since  0.66  +  0.34 
=  1. 


SUBTRACTION  19 

EXERCISE   5 

1.  Name  rapidly  the  complements  of  the  following 
numbers:  75,  64,  82,  12,  90,  33,  25,  0.25,  O.lG,  125,  500"^, 
5000,  1250,  625. 

2.  Name  the  amount  of  change  a  clerk  must  return  if 
he  receives  a  five-dollar  bill  in  payment  of  each  of  the 
following  amounts:  il.25,  ^3.75,  $^2.34,  83.67,  10.25, 
$0.88,  -S4.91,  11.85. 

3.  Name  the  amount  of  change  returned  if  the  clerk 
receives  a  ten-dollar  bill  in  payment  of  each  of  the  follow- 
ing amounts:  17.34,  $3.42,  $9.67,  15.25,  12.67,  $6.45, 
$4.87,  $0.68,  $3.34. 

Determine  in  each  of  the  following  exercises  what  num- 
ber added  to  the  smaller  number  will  produce  the  larger. 
The  student  will  notice  that  in  some  cases  the  subtrahend 
is  placed  over  the  minuend.  It  is  often  convenient  in 
business  to  perform  work  in  this  wa}^ 

4.  5.  6.  7.  8.  9. 

9      36      75      246      8937      5280 
5      42      31      167      9325      3455 


10.  11.  12.  13. 

7621  2339  9654327  4680215 

6042  5267  6098715  9753142 


14.  Show  that  to  subtract  73854  from  100000  it  is 
necessary  only  to  take  4  from  10  and  eacli  of  the  remain- 
ing digits  from  9. 

15.  Subtract  76495  from  100000,  and  397.82  from  1000, 
as  in  Ex.  14. 


20 


SUBTRACTION 


6,    9,  15  and  2;  17. 

5,  12,  15  and  0;  15. 

4,    6,    8  and  9  ;  17. 

4,    6,    7  and  1 ;  8. 


16.  Show  that  to  subtract  3642  from  5623  is  the  same 
as  to  add  the  arithmetical  complement  of  3642  and  sub- 
tract 10000  from  the  sum. 

17.  From  8757  take  the  sum  of  1236,  2273  and  3346. 

8757 
1236 
2273 
3346 
1902 

18.  From  53479  take  the  sum  of  23,  1876  and  41253. 

19.  From  7654  take  the  sum  of  3121,  126  and  2349. 

20.  From  764295  take  the  sum  of  45635,  67843, 125960 
and  213075. 

21.  A  clerk  receives  a  twenty-dollar  bill  in  payment  of 
the  following  items:  12.25,  111.50,  10.13,  10.75.  How 
much  change  does  he  return  ? 

22.  Find  the  value  of  2674  +  1782  -  1236  +  8420-4536 
by  adding  the  proper  arithmetical  complements  and  sub- 
tracting the  proper  powers  of  ten. 


30.    To  find  the  balance  of  an  account. 
Dr.    First  National  Bank,  Ypsilanti,  in  acct.  with  John  Smith     Cr. 


1904 

1904 

Aug.  3 

Balance 

1 

486 

87 

Aug.  4 

By  check 

500 

00 

Aug.  22 

To  deposit 

290 

00 

Aug.  10 

By  check 

57 

30 

Sept.  30 

To  deposit 

198 

75 

Sept.  1 

By  check 

235 

75 

Oct.  24 

To  deposit 

773 

40 

Sept.  21 

By  check 

11 

80 

Nov.  20 

To  deposit 

110 

Oct.  15 

By  check 

97 

30 

Nov.  3 

By  check 

1 

000 

00 

Nov.  25 

Balance 

956 

87 

2 

859 

02 

2 

859 

02 

Nov.  25 

Balance 

O.^P) 

87 

SUBTRACTION 


21 


The  preceding  form  represents  the  account  of  Jolm  Smith  with  the 
First  National  Bank  from  Aug.  3  till  Nov.  25.  The  items  at  the  left 
of  the  central  dividing  line  are  the  amounts  that  the  bank  owes  Mr. 
Smith.  This  side  is  called  the  debit  side  of  the  account.  The  items 
at  the  right  represent  the  amounts  withdrawn  by  ]\Ir.  Smith.  This 
side  is  called  the  credit  side  of  the  account.  The  diiference  y)etvveen 
the  sums  of  the  credits  and  the  debits  is  called  the  balance  oi  the 
account. 

It  is  evident  that  the  debit  side  of  the  above  account  is  greater 
than  the  credit  side.  Therefore,  to  balance  the  account,  add  the 
debit  side  first,  and  then  subtract  the  sum  of  the  credit  side  from  the 
result,  as  in  Ex.  17  above.  The  difference  will  be  the  balance,  or  the 
amount  left  in  the  bank  to  the  credit  of  ]Mr.  Smith.  The  work  can 
be  checked  by  adding  the  balance  to  the  credit  column.  The  result 
should  equal  the  sum  of  the  debit  column. 


EXERCISE  6 

Find  the  balance  of 

each  of  the  following 

accounts 

: 

1. 

Dr.             Cr. 

2. 

Dr.             Cr. 

3. 

Dr. 

Cr., 

234 

50 

246 

84 

798 

34 

125 

00 

500 

00 

450 

00 

212 

60 

55 

30 

351 

00 

97 

30 

100 

00 

60 

00 

75 

00 

198 

30 

1250 

00 

527 

30 

888 

80 

131 

60 

210 

60 

927 

50 

1100 

00 

500 

00 

2681 

50 

975 

25 

69 

00 

659 

75 

10 

46 

50 

00 

100 

00 

235 

67 

564 

90 

1000 

00  ! 

75 

00 

750 

25 

34 

68 

104 

69 

100 

00 

566 

66 

1200 

00 

195 

75 

275 

80 

302 

00 

625 

30 

259 

00 

4.  On  May  1  R.  F.  Joy  had  a  balance  of  f  1376.2-t  to 
his  account  in  the  bank.  He  deposited  on  May  1,  ^189; 
June  27,  1166;  July  28,  175;  Aug.  5,  $190.60;  Aug.  10, 
i  192.22.  He  withdrcAv  by  check  the  following  amounts  : 
June  1,  1153;  June  10,  1300;  July  3,  #25;  July  27, 
1575.50.     What  was  his  balance  Aug.  15? 


MULTIPLICATION 

31.  The  multiplication  table  should  be  so  well  known 
that  the  factors  will  at  once  suggest  the  product.  Thus, 
7  X  6,  or  6  X  7,  should  at  once  suggest  42. 

32.  The  student  should  also  be  able  to  see  at  once  what 
number  added  to  the  product  of  two  numbers  will  produce 
a  given  number.  Thus,  the  number  added  to  4  x  9  to 
produce  41  is  5,  or  4  x  9  and  5  are  41. 

It  is  a  common  practice  in  multipHcation  to  write  the  multiplier 
first  as  2  X  I  5  =  ^  10.  In  this  case  the  sign  (  x  )  is  read  "  times." 
If  the  multiplier  is  written  after  the  multiplicand,  as  in  $5  x  2  =  $10, 
the  sign  (x)  is  read  "multiplied  by."  The  multiplier  is  always  an 
abstract  quantity  (Why?),  but  the  multiplicand  may  be  either  abstract 
or  concrete. 

33.  The  following  examples  show  the  complete  process 
of  multiplication : 

Ex.  1.    Multiply  2743  by  356. 

Solution.     In    multiplying   one   number  by  2743  2743 

another  it  is  not  necessary  to  begin  with  the   ■         356  356 

units'  digit  of  the  multiplier.  We  may  begin  164.58 
with  either  the  units'  digit  or  the  digit  of  the  13715 
highest  order.  In  fact,  it  is  frequently  of  de-  §229 
cided  advantage  to  begin  with  the  digit  of  976,508 
highest  order,  especially  in  multiplying  deci- 
mals;  but  care  should  be  taken  in  placing  the  right-hand  figure  of 
the  first  partial  product.  Since  3  hundred  times  3  units  =  9  hundred, 
the  9  must  be  put  in  the  third  or  hundreds'  place,  etc. 


MULTIPLICATION  23 

Rv.  2.    Multiply  3.1416  by  213.34. 

Solution.     In  beginning  the  multiplicatiou  we  see  that 

20x0.0000  =  0.012.     Hence  the  2  is  written  in  the  thou-  •^■HIC 

sandths'  place.      The  work  is  then  completed  as  indicated  ^^^-'^^ 

in  the  annexed  example.     It  will  readily  be  seen  that  the  02.832 

rest  follows  after  pointing  off  the  first  partial   product  18.8496 

correctly.  .91:218 

The  advantage  of  beginning  with   the  digit  of   the  .12.5064 

highest   order    is   seen    in    approximations   (see   p.    59),  o.>  71 0-4.4. 
where  considerable  work  is  thereby  saved. 

34.    Check.      Multiplication   may  be   checked  by  usijig 
the   multiplicand   as   the   multiplier   and    performing   the 

multiplication  again.      However,   the  check  by  "casting 
out  the  nines "   (p.  41),  is  more  convenient. 


EXERCISE   7 

1.  Define  multiplier^  multiplicand,  product. 

2.  Explain  why  multiplication  is  but  an  abridged  method 
of  addition. 

3.  Can  the  multiplier  ever  be  a  concrete  number  ? 
Explain. 

4.  How  should  the  terms  be  arranged  in  multiplication  ? 
Does  it  make  any  difference  in  what  order  we  multiply  by 
the  digits  of  the  multiplier  ?  flight  we  begin  to  multiply 
with  the  5  in  Ux.  1  and  with  the  6  in  Ux.  2  ? 

5.  How  is  the  order  of  the  right-hand  figure  of  each 
partial  product  determined  ? 

6.  How  does  the  2^1'esence  of  a  zero  in  the  multiplier 
affect  the  work  ? 

7.  In  multiplying  3.1416  by  26.34,  can  we  tell  at  once 
how  many  integral  places  there  will  be  in  the  product  ? 
Can  we  tell  the  number  of  decimal  places  ? 


24  MULTIPLICATION 

8.  How  many  decimal  places  will  there  be  in  each  of 
the  following  products  :  21.34  x  5.9  ?  98.65  x  76.43  ? 
321.1  X  987.543  ?     1.438  x  42.345  ? 

35.    The  following  short  methods  are  useful : 

1.  To  multiply  any  numher  hy  5,  25,  16 J,  33|^,  125. 

Since  5  =  y-,  to  annex  a  cipher  and  divide  by  2  is  the  same  as  to 
multiply  by  5.  The  student  in  a  similar  manner  should  explain  short 
processes  of  multiplying  by  25,  16 1,  33|^,  125. 

2.  To  multiply  any  numher  hy  9. 

Since  9  =  10  —  1,  it  is  sufficient  to  annex  a  cipher  to  the  number 
and  subtract  the  original  number. 

Ex.    Multiply  432  by  9. 

432  X  10  =  4320 
432  X    1  =    432 


432  X    9  =  3888 


3.    To  multiply  any  7imnher  hy  11. 

Since  11  =  10  -f  1,  it  is  sufficient  to  annex  a  cipher  to  the  number 
and  add  the  original  number. 

Ex.    Multiply  237  by  11. 


237  X  10  =  2370 
237  X    1  =    237 


237  X  11  =  2607 

This  result  can  readily  be  obtained  by  writing  down  the  right-hand 
figure  first  and  then  the  sums  of  the  first  and  second  figures,  the  sec- 
ond and  third,  etc.,  and  finally  the  left-hand  figure. 

4.  To  multiply  any  mmiher  hy  a  7iumher  differing  hut 
little  from  soine  poioer  of  10. 

Annex  as  many  ciphers  to  the  number  as  there  are  ciphers  in  the 
next  higher  power  of  10,  and  subtract  the  product  of  the  number 
multiplied  by  the  complement  of  the  multiplier. 


MULTIPLICATION  25 

Ex.    Multiply  335  by  996.  900  =  1000  -  4. 

335  X  1000  =  33r)000         In  practice  written  335 
335  X      .  4  =      1310  1»M0 

335  X    996  =  333G0O  333(j(j0 

5.  To  midtiply  any  number  hy  a  number  of  two  fiyures 
ending  ivith  1. 

Multiply  by  the  tens'  figure  of  the  multiplier,  writing  this  product 
under  the  number  one  place  to  the  left. 

Ex.    Multiply  245  by  71. 

245  X  1  =  245 
245  X  70  =  17150 
245  X  71  =  17395 

6.  To  multiply  any  iiumher  hy  a  number  between  tivelve 
and  twenty. 

Multiply  by  the  units'  figure  of  the  multiplier,  writing  the  product 
under  the  number  one  place  to  the  right. 

Ex.    Multiply  427  by  13. 

427  X  10  =  4270 
427  X  3  .:=  1281 
427  X  13  =  5551 

7.  To  square  a  number  ending  hi  5. 

352  =  3  X  400  +  25,  452  z=  4  X  500  +  25,  55--2  =  5  x  600  +  25,  etc. 

8.  To  midtiply  by  a  number  when  the  multiplier  contains 
digits  ivhich  are  factors  of  other  parts  of  the  multiplier. 

Ex.    Multiply  25631  by  74221. 

Since  7  is  a  factor  of  42  and  21,  mnltiply  by  7.  placing  25631 

the  first  figure  in  the  partial  product  under  7.     (Why  V)  "4001 

Then  multiply  this  product  by  6  (42  :=  6  x  7),  placing  

the  first  figure  under  2  in  hundreds'  place.     (Why?)  I'-p-^.o 

Then  multiply  the  first  partial  product  by  3  (21  =  3x7),  ^^*  ^o .0-, 

placing  the  first  figure  under  1.     (Why?)     The  suui  of      — — II- 

these  partial  products  will  be  the  product  of  the  numbers.  190L3ob4ol 


26  MUL  TIPLICA  TION 

EXERCISE   8 

Name  rapidly  the  products   of  the  successive  pairs  of 
digits  in  each  of  the  following  numbers : 

1.  75849374657.  3.   67452367885. 

2.  265374867598.  4.   98765432345. 

5.  In  each  of  the  following  groups  of  digits  add  rapidly 
to  the  product  of  the  first  two  the  sum  of  all  that  follow : 
567,  432,  7654,  3456,  9753,  3579,  8642,  2468,  7896,  5436, 
3467. 

6.  Multiply  1264  by  125 ;  by  121 ;  by  IJ. 

7.  Multiply  76.26  by  16f  ;  by  331 

8.  Multiply  2348  by  25;  by  21;  by  50;  by  0.5. 

9.  Multiply  645  by  9;  by  11 ;  by  17  ;  by  41. 

10.  Multiply  8963  by  848. 

11.  Multiply  37439  by  4832. 

12.  Show  that  to  multiply  a  number  by  625  is  the  same 
as  to  multiply  by  10000  and  divide  by  16. 

13.  Subtract  5  x  12631  from  87642. 
The  work  should  be  done  as  follows : 


5x1  and  7,  12. 

87642 

5  X  3  and  1  and  8, 

24. 

12631 

5x6  and  2  and  4, 

36. 

24487 

5x2  and  3  and  4, 

17. 

5x1  and  1  and  2, 

8. 

14.  Subtract  3  x  2462  from  9126. 

15.  Subtract  6  x  42641  from  768345. 


MULTIPLICATION 


27 


1 

784 

2 

1568 

'.] 

2:i52 

4 

ai;J6 

5 

3920 

6 

4704 

7 

5488 

8 

6272 

9 

7056 

16.  Subtract  2  x  8(1478  from  291872. 

When  the  same  miinher  is  to  be  used  as  a  tiiultii)lier  several  times, 
work  may  be  saved  l>y  t'oruiiiig  a  table  of  its  multiples.     'J'lius, 

5764  X  784  = 

3i:]6  (4) 

4704  (6) 

5488  (7) 

3920  (5) 

4518976 

The  partial  products  in  each  case  are 
taken  from  the  table. 

17.  Use  the  above  table  and  multiply  5764,  74591,  84327, 
23145,  each  by  784. 

18.  Form  a  table  of  multiples  of  6387,  and  use  it  to  find 
the  product  of  7482,  3.1416,  742896,  342312,  67564584, 

897867,  65768798,  56024.85,  each  by  6387. 

19.  Multiply  2785  by  9998,  and  1728  by  997. 

20.  Multiply  78436  by  25  x  125. 

21.  Multiply  32.622  by  0.0125. 

22.  Multiply  486.72  by  0.25  x  0.25. 

23.  Multiply  320.4  by  5  X  1.25. 

24.  Multiply  5763  by  16f  x  33f     ' 


DIVISION 

36.  In  division  the  student  should  be  able  to  see  at 
once  how  many  times  a  given  digit  is  contained  in  any 
number  of  two  digits  with  the  remainder.  Thus,  7  is 
contained  in  46,  6  times  with  a  remainder  4.  The  student 
sliould  think  simply  6  and  4  over. 

Ex.    6)354279 

59046  remainder  3. 

The  whole  mental  process  should  be  5  and  5,  9  and  0,  0  and  2, 
4  and  3,  6  and  3. 

Two  interpretations  arise  from  considering  division  as  the  inverse 
of  multiplication. 

Thus,  since  4  x  <|6  =  ,|24. 

(1)  $24  ^  4  =  |6,  separation  into  groups.  $24  has  been  separated 
into  4  equal  groups. 

(2)  $24 -^$6  =  4,  involving  the  idea  of  measuring,  or  being  con- 
tained in. 

$6  is  contained  in  $24,  4  times. 

37.  The  following  examples  show  the  complete  process 
of  long  division. 

346 

4541)1571186 

1^6-'^  It  assists  in   determining  the  order 

20888  ^^  ^'^^  digits  in  the  quotient  to  write 

18164  them  in  their  proper  places  above  the 

dividend. 

27246 

27246 

28 


DIVISION  29 

38.  The  work  in  long  division  may  be  very  much 
abridged  by  omitting  the  partial  products  and  writing 
down  the  remainders  only.  These  remainders  are  ob- 
tained by  the  method  used  in  Ex.  13,  p.  26. 

Ux,  Divide  764.23  by  2.132. 

The  work  will  be  simplified  by  multiplying  both  niimljers  by  1000 
to  avoid  decimals.     The  first  remainder,  124(),  is  obtained  as  follows  : 

358 

2132)764230  3  x  2,  0  and  6,  12. 

12463  3  X  3,  9  and  1,  10  and  4,  14. 

18030  3  X  1,  3  and  1,  4  and  2,  6. 

974  3  X  2,  6  and  1,  7. 

Then  bring  down  3  and  proceed  as  before  to  form  the  other 
remainders. 

39.  Check.  Division  may  be  checked  by  multiplying 
the  quotient  by  the  divisor,  the  product  plus  the  remainder 
should  equal  the  dividend.  The  check  by  ''casting  out 
the  nines  "  (p.  42)  may  be  used. 

EXERCISE   9 

1.  Define  dividend^  divisor,  quotient^  remainder. 

2.  Explain  the  two  interpretations  arising  from  consider- 
ing division  as  the  inverse  of  multiplication.  5  x  8 10  =  "5^  b'^. 
Give  the  two  interpretations  as  applied  to  this  example. 

3.  How  is  the  order  of  the  right-hand  figure  in  each 
partial  product  determined  ? 

4.  Explain  why  the  sum  of  the  partial  products  plus  the 
remainder,  if  any,  must  equal  the  dividend  if  the  w^ork  is 
correct. 

5.  Explain  why  the  quotient  is  not  affected  by  multi- 
plying both  dividend  and  divisor  by  the  same  number. 


30 


DIVISION 


40.  If  the  same  number  is  used  as  a  divisor  several 
times,  or  if  the  dividend  contains  a  large  number  of 
places,  work  may  be  saved  by  forming  a  table  of  multiples 
of  the  divisor.     Thus  : 

Ux.    Divide  786342  by  4147. 


1 

4147 

189 

2 

8294 

4147) 

786342 

3 

12441 

37164 

4 

16588 

39882 

5 

20735 

2559  remainder 

6 

24882 

7 

29029 

8 

38176 

9 

37323 

EXERCISE    10 

1.  Divide  987262,  49789  and  314125  each  by  4147. 

2.  Divide  896423,  76425,  9737894  each  by  5280. 

3.  Divide  44.2778  by  63.342. 

Find  the  value  of  : 

4.  32.36-8.9. 

5.  1.25 -^  0.5  and  12.5-0.05. 

6.  144-1.2  and  14.4-12. 

7.  625-25  and  62.5^2.5. 

8.  1125-50  and  11.25 -j- 9.5. 

9.  5280-12.5  and  580-125. 
10.  750-2.5^0.5. 

41.  In  addition  to  the  checks  on  the  fundamental  pro- 
cesses given  above,  it  is  well  wlien  possible  to  foryn  the 
habit  of  estimating  results  before  beginning  the  solution  of  a 


DIVISION  31 

problem.     Thus,  in  iiiultijjlyiiig  11*.]  by  12]   it  is  evident 
that  the  result  will  be  about  12  x  20=  240. 

In  using  this  check  the  student  should  form  a  rough 
estimate  of  the  result,  then  solve  the  problem  and  com- 
pare results.     A  large  error  will  be  at  once  detected. 

EXERCISE    11 

Solve  the  following,  first  giving  approximate  answers, 
then  the  correct  result  : 

1.  Multiply  15.3  x  3|f  (about  15  x  4). 

2.  Divide  594  by  -f^^  (about  594  -  i). 

3.  Divide  32.041  by  0.499  (about  32.041  -  i). 

4.  How  much  will  21  horses  cost  at  S145  each? 

5.  Multiply  30.421  by  20.516. 

6.  At  12 J  ct.   a  dozen,  how  much  will  Q^  doz.  eggs 
cost  ? 

7.  At  37 1  ct.  a  pound,  how  much  will  11  lb.  of  coffee 
cost  ? 

8.  How  many  bushels  of  potatoes  can  be  bought  for 
$5.25  at  35  ct.  a  bushel? 

9.  At  $1,121   a  barrel,  how  many  barrels  of  salt  can 
be  bought  for  122.50? 

10.  How  far  will  a  train  travel  in  121  ]n\  at  the  rate  of 
45  mi.  an  hour  ? 

11.  How  much  will  8|  T.  of  coal  cost  at  $7.25  a  ton  ? 

12.  The  net  cost  of  printing  a  certain  book  is  49  ct.  a 
copy.     How  much  will  an  edition  of  2500  cost  ? 

13.  At  the  rate  of  40  mi.  an  hour,  how  long  will  it 
take  a  train  to  run  285  mi.  ? 


FACTORS   AND   MULTIPLES 

42.  A  factor  or  divisor  of  a  number  is  any  integral 
number  that  will  exactly  divide  it. 

43.  A  number  that  is  divisible  by  2  is  called  an  even 
number,  and  one  that  is  not  divisible  by  2  an  odd  number. 

Thus,  24  and  58  are  even  numbers,  while  17  and  83  are  odd  numbers. 

44.  A  number  that  has  no  factors  except  itself  and 
unity  is  called  a  prime  number. 

Thus,  1,  2,  3,  5,  7,  etc.,  are  prime  numbers. 

45.  Write  down  all  of  the  odd  numbers  less  than  100 
and  greater  than  3.  Beginning  with  3  reject  every  third 
number  ;  beginning  with  5  reject  every  fifth  number ; 
beginning  with  7  reject  every  seventh  number.  The 
numbers  remaining  will  be  all  of  the  prime  numbers 
between  3  and  100.     (Why?) 

46.  This  method  of  distinguishing  prime  numbers  is  called  the 
Sieve  of  Eratosthenes,  from  the  name  of  its  inventor,  Eratosthenes 
(276-196  B.C.).  He  wrote  the  numbers  on  a  parchment  and  cut  out 
the  composite  numbers,  thus  forming  a  sieve. 

47.  A  number  that  has  other  factors  besides  itself  and 
unity  is  called  a  composite  number. 

48.  Numbers  are  said  to  be  prime  to  each  other  when  no 
number  greater  than  1  will  exactly  divide  each  of  them. 

Are  numbers  that  are  prime  to  each  other  necessarily 
prime  numbers  ? 


FACTORS  AND  MULTIPLES  33 

49.  An  integfral  nninbcr  tluit  will  exactly  divide  two  or 
more  numbers  is  called  a  common  divisor,  or  a  common 
factor  of  these,  numbers. 

Thus,  2  and  ^3  are  conunon  divisors  of  12  and  18. 

50.  The  greatest  common  factor  of  two  or  more  num- 
bers is  called  the  greatest  common  divisor  (g.  c.  d.)  of  the 
numbers. 

Thus,  6  is  the  g.  c.  d.  of  12  and  18. 

51.  A  common  multiple  of  two  or  more  numbers  is  a 
number  that  is  exactly  divisible  by  each  of  them. 

Thus,  12,  18,  24,  aud  48  are  common  multiples  of  3  and  6,  while  12 
is  the  least  common  multiple  (1.  c.  m.)  of  3  and  G. 

52.  It  is  of  considerable  importance  in  certain  arith- 
metical operations,  particularly  in  cancellation,  to  be  able 
readily  to  detect  small  factors  of  numbers.  In  proving 
the  tests  of  divisibility  by  such  factors,  the  two  following 
principles  are  important. 

1.  A  factor  of  a  numher  is  a  factor  of  any  of  its 
multiples. 

Proof.     Every  multiple  of  a  number  contains  that  number  an  exact 
number  of  times;  therefore,  it  contains  every  factor  of  tlie  number. 
Thus,  5  is  a  factor  of  25,  and  hence  of  3  x  25,  or  75. 

2.  A  factor  of  any  ttvo  numhers  is  a  factor  of  the  sum  or 
difference  of  any  two  multiples  of  the  numhers. 

Proof.  Any  factor  of  two  numbers  is  a  factor  of  any  of  their  mul- 
tiples by  Principle  1.  Therefore,  as  each  nudtiple  is  made  up  of  parts 
each  equal  to  the  given  factor,  their  sum  or  difference  will  be  made  up 
of  parts  equal  to  the  given  factor,  or  will  be  a  multiple  of  the  given 
factor. 

Thus,  3  is  a  factor  of  12  and  of  15,  and  hence  of  5  x  12  +  2  x  15, 
or  90.     3  is  also  a  factor  of  5  x  12  —  2  x  15,  or  30. 
ltman's  adv.  ar.  — 3 


34  FACTORS  AND  MULTIPLES 

53.  Tests  of  Divisibility.  1.  An^  number  is  divisible  by 
2  if  the  number  represented  by  its  last  right-hand  digit  is 
divisible  by  2. 

Proof.  Any  number  may  be  considered  as  made  up  of  as  many 
lO's  as  are  represented  by  the  number  exclusive  of  its  last  digit  plus  the 
last  digit.  Then,  since  10  is  divisible  by  2,  the  first  part,  which  is  a 
multiple  of  10,  is  divisible  by  2.  Therefore,  if  the  second  part,  or  the 
number  represented  by  the  last  digit,  is  divisible  by  2,  the  whole 
number  is. 

Thus,  634  =  63  X  10  +  4  is  divisible  by  2  since  4  is. 

2.  Any  number  is  divisible  by  4  if  the  number  represented 
by  the  last  two  digits  is  divisible  by  4. 

Proof.  Any  number  may  be  considered  as  made  up  of  as  many 
lOO's  as  are  represented  by  the  number  exclusive  of  its  last  two  digits 
plus  the  number  represented  by  the  last  two  digits.  Then,  since  100  is 
divisible  by  4,  the  first  part,  which  is  a  multiple  of  100,  is  divisible 
by  4.  Therefore,  if  the  number  represented  by  the  last  two  digits  is 
divisible  by  4,  the  whole  number  is. 

Thus,  85648  =  856  x  100  +  48  is  divisible  by  4  since  48  is. 

3.  Any  nu?nber  is  divisible  by  5  if  the  last  digit  is  0  or  5. 
The  proof,  which  is  similar  to  the  proof  of  1,  is  left  for  the  student. 
Note.     0  is  divisible  by  any  number,  and  the  quotient  is  always  0. 

4.  Any  number  is  divisible  by  8  if  the  number  represented 
by  its  last  three  digits  is  divisible  by  8. 

The  proof  is  left  for  the  student. 

5.  Any  number  is  divisible  by  9  if  the  sum  of  its  digits  is 
divisible  by  9. 

Proof.  Since  10  =  9  +  1,  any  number  of  lO's  =  the  same  number 
of  9's  +  the  same  number  of  units;  since  100=  99  +  1,  any  number 
of  lOO's  =  the  same  number  of  99's  +  the  same  number  of  units  ;  since 


FACTORS  AND   MULTIPLES       '  35 

1000  =  9f)f)  +  1,  any  iiiinil)er  of  lOOO's  =  the  same  nuin})er  of  ODO's 
+  the  same  number  of  units;  etc.  Therefore,  any  number  is  made  up 
of  a  multiple  of  9  +  the  sum  of  its  digits,  and  hence  is  divisil^le  by  9 
if  the  sum  of  its  digits  is  divisible  by  9. 

Thus,  7:]Gl>  =  7  x  1000  +  8  x  100  +  G  x  10  +  2 

=  7(999  +  1)  +  ;5(99  +  1 )  +  0(9  +  1)  +  2 

=  7  X  999  +  8  X  99  +  6x9  +  7  + 13 +  6+2 

=  a  multiple  of  9  +  the  sum  of  the  digits. 

Therefore,  the  number  is  divisible  by  9  since  7  +  3  +  6  +  2  =  18  is 
divisible  by  9. 

6.  Any  number  is  divisible  by  3  if  the  sum  of  its  digits  is 
divisible  by  3. 

The  proof,  which  is  similar  to  the  proof  of  Principle  .5,  is  left  for  the 
student. 

7.  Any  even  number  is  divisible  by  6  if  the  sum  of  its  digits 
is  divisible  by  3. 

The  proof  is  left  for  the  student. 

8.  A7iy  number  is  divisible  by  11  if  the  difference  between 
the  sums  of  the  odd  and  even  orders  of  digits,  counting  from 
units^  is  divisible  by  11. 

Proof.  Since  10  =  11  —  1,  any  number  of  lO's  =  the  same  number 
of  ll's—  the  same  number  of  units;  since  100  =  99+1,  any  number  of 
lOO's  =  the  same  number  of  99's  +  the  same  number  of  units ;  since 
1000  =  1001  -  1,  any  number  of  lOOO's  =  the  same  number  of  lOOl's 
—  the  same  number  of  units ;  etc.  Therefore,  any  number  is  made 
up  of  a  multiple  of  11  +  the  sum  of  the  digits  of  odd  order  —  the  sum 
of  the  digits  of  even  order,  and  hence  is  divisible  by  11  if  the  sum  of 
the  digits  of  odd  order  —  the  sum  of  the  digits  of  even  order  is  divis- 
ible bv  11. 


36  •     FACTORS  AND   MULTIPLES 

Thus,  753346  =  7x100000  +  5x10000  +  3x1000  +  3x100  +  4x10  +  6 

=  7(100001-1) +  5(9999  +  1)  + 3(1001-1) 

+  3(99  +  l)+4(ll-l)+6 

=  7x100001  +  5x9999  +  3x1001  +  3x99  +  4x11 

.  -7+5-3+3-4+6 

=  a  multiple  of  11  + the  sum  of  the  digits  of  odd  order 

—  the  sum  of  the  digits  of  even  order. 

Therefore,  the  number  is  divisible  by  11  since  5+3  +  G  —  (7  +  3  +  4)  =  0 
is  divisible  by  11. 

9.  The  test  for  divisihility  hy  1  is  too  complicated  to  he 
useful, 

EXERCISE    12 

1.  Write  three  numbers  of  at  least  four  figures  each 
that  are  divisible  by  4. 

2.  Write  three  numbers  of  six  figures  each  that  are 
divisible  by  9. 

3.  Is  352362257  divisible  by  11  ?   by  3  ? 

4.  Without  actual  division,  determine  what  numbers 
less  than  19  (except  7,  13,  14,  17)  will  divide  586080. 

5.  Explain  short  methods  of  division  by  5,  25,  16|, 
331  125. 

6.  Divide  3710  by  5 ;  by  25;  by  125;  by  121. 

7.  Divide  2530  by  0.5;  by  0.025;  by  1.25. 

8.  Prove  that  to  divide  by  625  is  the  same  as  to  mul- 
tiply by  16  and  divide  by  10000. 

9.  State  and  prove  a  test  for  divisibility  by  12 ;    by 
15;  by  18. 

10.  If  7647  is  divided  by  2  or  5,  how  will  the  remainder 
differ  from  the  remainder  arising  from  dividing  7  by  2  or 
5  ?     Explain. 


FACTORS   Ay  I)   MULTIPLES  37 

11.  If  2()72T  is  divided  l)y  4  oi*  2"),  liow  will  llic  reniiiiii- 
der  differ  from  the  remainder  arising-  from  dividing-  27  l^y 
4  or  25  ?     Explain . 

12.  Explain  how  you  can  hnd  the  remaindei-  iirising-  from 
dividing  26727  by  8  or  125  in  the  sliortest  possiljle  way. 

54.  Relative  Weight  of  Symbols  of  Operation.      In  the 

use  of  the  symbols  of  operation  (  +  ,  — ,  x  ,  ^),  it  is  impor- 
tant that  the  student  should  know  that  the  numbers  con- 
nected by  the  signs  x  and  -r-  must  first  be  operated  upon 
and  then  tliose  connected  by  +  and  —  ;  for  the  signs  of 
multiplication  and  division  connect  factors,  while  the  signs 
of  addition  and  subtraction  connect  terms.  Factors  must 
be  combined  into  simple  terms  before  the  terms  can  be 
added  or  subtracted. 

Thus,  5  +  2  X  3  -  15  --  5  +  4  =  12,  the  terms  2x3  and  15  -f-  5  being 
simplified  before  they  are  combined  by  addition  and  subtraction. 

55.  The  ancients  had  no  convenient  symbols  of  operation.  Addi- 
tion was  generally  indicated  by  placing  the  numbers  to  be  added  adja- 
cent to  each  other.  Other  operations  were  written  out  in  words.  The 
symbols  +  and  —  were  probably  first  used  by  Widman  in  his  arithmetic 
published  in  Leipzig  in  1489.  He  used  them  to  mark  excess  or  defi- 
ciency, but  they  soon  came  into  use  as  symbols  of  operation,  x  as  a 
symbol  of  multiplication  was  used  by  Oughtred  in  1G31.  The  dot  ( •  ) 
for  multiplication  was  used  by  Harriot  in  1631.  The  Arabs  indicated 
division  in  the  form  of  a  fraction  quite  earlv.  ^  as  a  symbol  of  divi- 
sion v;as  used  hy  Rahn  in  his  algeV)ra  in  1059*  Robert  Recorde  intro- 
duced the  symbol  =  for  equality  in  15.")7.  :  was  used  to  indicate 
division  by  Leibnitz  and  Clairaut.  In  1631  Harriot  used  >  and  < 
for  greater  than  and  less  than.  Rudolff  used  y/  to  denote  square  root 
in  1526. 

56.  Greatest  Common  Divisor.  In  many  cases  the  g.  c.  d. 
of  two  or  more  numbers  may  readily  be  found  by  factoring, 
as  in  the  following  example : 


377 

3 

348 

1 

29 

12 

38  FACTORS  AND  MULTIPLES 

Ex.     Find  the  g.  c.  d.  of  3795,  7095,  30030. 

3795  =  3  X  5  X  11  X  23, 

7095  =  3  X  5  X  11  X  43, 

30030  =  2  X  3  X  5  X  11  X  91, 

and  since  the  g.  c.  d.  is  the  product  of  all  of  the  prime  factors  that  are 
common  to  the  three  numbers,  it  is  3x5x11  =  165. 

57.  Euclid,  a  famous  Greek  geometer,  who  lived  about  300  B.C., 
gave  the  method  of  finding  the  g.  c.  d.  by  division.  This  method  is 
useful  if  the  prime  factors  of  the  numbers  cannot  be  readily  found. 

Ex.     Find  the  g.  c.  d.  of  377  and  1479. 

1479 

, .  o.  The  g.  c.  d.  cannot  be  greater  than  377,  and  since 

.3  ,Q       377  is  not  a  factor  of  1479,  it  is  not  the  g.  c.  d.  of  the 

oj^Q       two  numbers. 

Divide  1479  by  377.  Then,  since  the  g.  c.  d.  is  a  common  factor  of 
377  and  1479,  it  is  a  factor  of  1479-3  x  377,  or  348  (Principle  2,  p.  33). 

Therefore,  the  g.  c.  d.  is  not  greater  than  348.  If  348  is  a  factor  of 
377  and  1479,  it  is  the  g.  c.  d.  sought. 

But  348  is  not  a  factor  of  377.    Therefore,  it  is  not  the  g.  c.  d.  sought. 

Divide  377  by  348.  Then,  since  the  g.  c.  d.  is  a  factor  of  377  and 
348,  it  is  a  factor  of  377  -  348,  or  29  (Principle  2,  p.  33). 

Therefore,  the  g.  c.  d.  is  not  greater  than  29,  and  if  29  is  a  factor  of 
348,  377,  and  1479,  it  is  the  g.  c.  d.  sought.     (Why  ?) 

29  is  a  factor  of  348.  Therefore,  it  is  a  factor  of  377  and  of  1479. 
(Why?) 

Therefore,  29  is  the  g.  c.  d.  sought. 

58.  Least  Common  Multiple.  In  many  cases  the  1.  c.  m. 
of  two  or  more  numbers  may  readily  be  found  by  factoring, 
as  in  the  following  example. 

Ex.    Find  the  1.  c.  m.  of  414,  408,  3330. 
414  =  2  X  3  X  3  X  23, 

408  =  2  X  2  X  2  X  3  X  17, 
3330  =  2  X  3  X  3  X  5  X  37. 


FACTORS  AND  MULTIPLES  39 

The  ].c.  111.  must  contain  all  of  the  i)riine  factors  of  414,  408,  :>:}80, 
and  each  factor  must  occur  as  often  in  the  1.  c.  m.  as  in  any  one  of  the 
numbers.  Thus,  8  must  occur  twice  in  the  I.  c.  m.,  2  must  occur  three 
times,  and  28,  17,  -5,  87  must  each  occur  once. 

Therefore,  the  1.  c.  m.  =  2  x  2  x  2  x  3  x  8  x  5  x  28  x  17  x  87  = 
5208120. 

59.  When  the  numbers  cannot  readily  be  factored,  the 
g.  c.  d.  may  be  used  in  finding  the  1.  c.  m. 

Since  the  g.  c.  d.  contains  all  of  the  factors  that  are 
common  to  the  numbers,  if  the  numbers  are  divided  by 
the  g.  c.  d.,  the  quotients  will  contain  all  the  factors  that 
are  not  common.  The  l.c.m.  is  therefore  the  product  of 
the  quotients  and  the  g.  c.  d.  of  the  numbers. 

Ex.    Find  the  1.  c.  m.  of  14482  and  32721. 

The  g.  c.  d.  of  14182  and  82721  is  18. 

14:482  ^  18  =  1114:.     .-.  the  1.  c.  m.  of  the  two  numbers  is 

llUx  82721  =  86451194. 
EXERCISE   13 

1.  Find  the  1.  c.  m.  and  g.  c.  d.  of  384,  2112,  2496. 

2.  Find  the  1.  c.  m.  of  3,  5,  9,  12,  14,  16,  96,  128. 

3.  Find  the  g.  c.  d.  and  1.  c.  m.  of  1836,  1482,  1938, 
8398,  11704,  101080,  138945. 

4.  Prove  that  the  product  of  the  g.  c.  d.  and  1.  c.  m. 
of  two  numbers  is  equal  to  the  product  of  the  numbers. 

5.  What  is  the  length  of  the  longest  tape  measure  that 
can  be  used  to  measure  exactly  two  distances  of  2916  ft. 
and  3582  ft.  respectively  ? 

6.  Find  the  number  of  miles  in  the  radius  of  the  earth, 
having  given  that  it  is  the  least  number  that  is  divisible 
by  2,  3,4,  5,  6,  9,  10,  11,  12. 


CASTING   OUT   NINES 

60.  The  check  on  arithmetical  operations  by  casting 
out  the  nines  was  used  by  the  Arabs.  It  is  a  very  useful 
check,  but  fails  to  detect  such  errors  as  the  addition  of  9, 
the  interchange  of  digits,  and  all  errors  not  affecting  the 
sum  of  the  digits.      (Why?) 

The  remainder  cunsing  from  dividing  any  7iumber  hy  9  is 
the  same  as  that  arisijig  from  dividing  the  sum  of  its  digits 
Jy  9.  '  • 

Thus,  the  remainder  arising  by  dividing  75234:  by  9  is  3,  the  same 
as  arises  by  dividing  7  +  5  +  2  +  3  +  4  by  9. 

The  student  should  adapt  the  proof  of  Principle  5,  ix  34,  to  this 
statement. 

61.  The  most  convenient  method  is  to  add  the  digits, 
dropping  or  "  casting  out "  the  9  as  often  as  the  sum 
amounts  to  that  number. 

Thus,  to  determine  the  remainder  arising  from  dividing  645738  by 
9,  say  10  (reject  9),  1,  6,  13  (reject  9),  4,  7,  15  (reject  9),  6.  There- 
fore, 6  is  the  remainder.  After  a  httle  practice  the  student  will  easily 
group  the  9's.  In  the  above,  6  and  3,  4  and  5,  could  be  dropped,  and 
the  excess  in  7  and  8  is  seen  to  be  6  at  once. 

62.  Check  on  Addition  by  casting  out  the  9*s. 

Ux.  Add  56342,  64723,  57849,  23454  and  check  the 
work  by  casting  out  the  9's. 

40 


CASTING    OUT  NINES  41 

Since  each  number  is  a  nuiltipl(,'  of  9  plus  some  remaiiultT,  the 

numbers  can  be  written  as   indicated 
56342  =  9  X    0200  +    2  rem.     j,,  the  annexed  sohition. 
64723  =  9  X    719L+    4  rem. 

57849  =  9  X    0127+    6  rem.  But  12  =  9  +  3. 

23454  =  9  X    2000  +    0  rem.  .^  202308  =  9  x  22-184  +  9  +  3 

202308  =  9  X  22484  +  12  rem.  ^  ^  ^  ^2485  +  3. 

Thus,  the  excess  of  9's  is  3  and  the  excess  in  tlie  sum  of  the  ex- 
cesses, 2,  4,  6,  and  0,  is  3,  therefore  the  work  is  probably  correct. 


63.   The  proof  may  be  made  general  by  writing  the  numbers  in 

the  form   9  a:  +  r.      This  can    be  done  since  all 

',  ^    ,  numbers  are   multiples  of  9  plus  a  remainder. 

"    „  '     ,,  Hence,  by  expressing  the  numbers  in  this  form 

"_ and  adding  w^e  have  for  the  sum  a  multiple  of  9 

plus  the  sum  of  the  remainders.     Therefore,  the 

Z~.  '.        ~  ~  excess  of  the  9's  in  the  sum  is  equal  to  the  excess  in 

9(x  +  x'  +  x"  -\--")  ^  ^ 

,        ,,  .  the  sum  of  the  excesses. 

+  r  +  r'  +  r"+  •■•)  -^ 


64.    Check  on  Multiplication  by  casting  out  the  g's. 

Since  any  two  numbers  may  be  written  in  the  form  9  a:  +  ;-  and 

9  x'  +  r',    multiplying    9  x  +  r   by    9  x'  +  r', 

^x  -^r  ^re  have   ^1  xx'  +  Q(^x'r -\- xr')  + rr'.     From 

^  X  +  f'  this  it  is  evident  that  the  excess  of  9's  in 

9  xr'  +  rr'  the  product  arises  from  the  excess  in  vr'. 

81  xx'  +  9  x'r  Therefore,  the  excess  of  9's  in  any  product  is 

^1  xx' -\- Q(x'r -\- xr')-\-rr'     equal   to   the  excess  in   the  product  found  hij 

multiplying  the  excesses  of  the  factors  together. 

Ex.    Multiply  3764  by  456  and  clieck  by  casting  out 

^^®  ^'^-  3761  X  -150  =  1710384. 

The  excess  of  9's  in  3764  is  2 ;  the  excess  in  456  is  6  ;  the  excess  in 
the  product  of  the  excesses  is  3  (2  x  6  =  12  ;  12  —  9  =  3);  the  excess 
in  1710381,  the  product  of  the  numbers,  is  3.  Therefore,  the  work  is 
probably  correct. 


42  CASTING   OUT  NINES 

65.    Check  on  Division  by  casting  out  the  p's. 

Division  being  tlie  inverse  of  multiplication,  the  dividend 
is  equal  to  the  product  of  the  divisor  and  quotient  plus  the 
remainder.  Therefore,  the  excess  of  9's  in  the  dividend  is 
equal  to  the  excess  of  9's  i7i  the  remainder  plus  the  excess 
in  the  product  found  by  multiplying  the  excess  of  9's  in  the 
divisor  by  the  excess  of  9's  in  the  quotient. 

Ex.     Divide   7456r3  by  428  and  check  by  casting  out 

^^^®   ^'^-  74563 -f- 428  =  174  + /2V 

or  74563  =  174  x  428  +  91 

The  excess  of  9's  in  74563  is  7  ;  in  174,  3  ;  in  428,  5 ;  in  91, 1.  Since 
7,  the  excess  of  9's  in  74563  =  the  excess  in  3  x  5  +  1,  or  16,  which  is 
the  product  of  the  excesses  in  174  and  428  plus  the  excess  in  91,  the 
work  is  probably  correct. 

EXERCISE   14 

1.  State  and  prove  the  check  on  subtraction  by  casting 
out  the  9's. 

2.  Determine  without  adding  whether  89770  is  the 
sum  of  37634  and  52146. 

3.  Add  74632,  41236,  897321  and  124762,  and  check 
by  casting  out  the  9's. 

4.  Multiply  76428  by  5937,  and  check  by  casting  out 
the  9's. 

5.  Determine  without  multiplying  wliether  2718895  is 
the  product  of  3785  and  721. 

6.  Show  by  casting  out  9's  that  18149  divided  by 
56  =  324^V 

7.  Show  that  results  may  also  be  checked  by  casting 
out  3's  ;   by  casting  out  ll's. 


MisCELLAXEOrs    EXEIiCISE  43 

8.  Is  734(;r)7  divisible  ])y  9?  by  3?  by  11? 

9.  lY'i'forin  till'  following  operations  and  clieck  : 
91728  X  762  ;  849631  -  2463  ;  17  x  3.1416  ;  78.04  -  3.1416. 

10.  Does  the  proof  for  casting  out  the  9\s  hold  as  well 
for  4,  6,  8,  etc.  ?  May  we  check  by  casting  out  the  8's '/ 
Explain. 

MISCELLANEOUS   EXERCISE   15 

1.  What  is  the  principle  by  which  the  ten  symbols,  1, 
2,  3,  4,  5,  6,  7,  8,  9,  0,  are  used  to  represent  any  number  ? 

2.  Why  is  the  value  of  a  number  unaltered  by  annexing 
zeros  to  the  right  of  a  decimal  ? 

3.  How  is  the  value  of  each  of  the  digits  in  the  number 
326  affected  by  annexing  a  number,  as  4,  to  the  right  of 
it  ?  to  the  left  of  it  ? 

4.  How  is  the  value  of  each  of  the  digits  of  7642 
affected  if  5  is  inserted  between  6  and  4  ? 

5.  Write  4  numbers  of  4  places  each  that  are  divisible 
by  ia)  4,  (5)  2  and  5,  (c)  6,  (c?)  8,  (^  9,  (/)  11,  Qg)  16, 
(A)  12,  (0  15,  (y)  18^A-)  3,  (0  50,  (m)  125,  (n),  both 
6  and  9,  (o)  both  8  alPfe,  (p)  both  30  and  20. 

6.  Determine  the  prime  factors  of  the  following  num- 
bers :  (a)  3426,  (5)  8912,  (c)  6600,  (i)  6534,  (^  136125, 
(/)  330330,  (^)  570240. 

7.  Mr.  Long's  cash  balance  in  the  bank  on  Feb.  20  is 
1765.75.  He  deposits,  Feb.  21,  .fl50;  P^eb.  25,  -^350.25; 
Feb.  26,  $97.50;  and  withdraws,  Feb.  23,  ^f  200;  Feb.  24, 
!?^123.40  and  .^112.50;  Feb.  28,  1321.75.  What  is  his 
balance  March  1  ? 


44  MISCELLANEOUS  EXERCISE 

8.  Form   a  table   of  multii)les   of  tlie  multiplier  and 
multiply  7642,  98856,  24245,  6420246,  each  by  463. 

9.  Form  a  table  of  multiples  of  tlie  divisor  and  use  it 
in  dividing  86420,  97531,  876123,  64208,  each  by  765. 

10.  Use  a  short  method  to  multiply  8426  by  16|  ;  by 
331 ;  by  945  ;  by  432. 

11.  Find,  without  dividing,  the  remainder  when  374265 
is  divided  by  3  ;   by  9  ;  by  11. 

12.  Evaluate  45  +  32  x  25  -  800  -  125  +  180  x  33i. 

13.  Determine  by  casting  out  the  9's  whether  the 
following  are  correct  :  (a)  786  x  648  =  509328  ;  (b) 
24486  -  192  =  127  +  102  rem.  ;  {c)  415372  -  267  = 
1555  +  187  rem. ;   (d)  16734  x  3081  =  52557454. 

14.  Perform  each  of  the  operations  indicated  in  Ex.  13. 

15.  Subtract  from  784236  the  sum  of  7834,  5286,  23462 
and  345679. 

16.  What  are  the  arithmetical  complements  of  12000, 
1728,  3.429,  86,  0.1,  125? 

17.  Light  travels  at  the  rate  of  186000  mi.  per  second. 
Find  the  distance  of  the  sun  from  the  earth  if  it  takes  a 
ray  of  light  from  the  sun  8  min.  2ijfec.  to  reach  the  earth. 

18.  A  cannon  is  2  mi.  distant  from  an  observer.  How 
long  after  it  is  fired  does  it  take  the  sound  to  reach  the 
observer  if  sound  travels  1090  ft.  per  second  ? 

19.  Replace  the  zeros  in  the  number  760530091  by 
digits  so  that  the  number  will  be  divisible  by  both  9  and  11. 

20.  Show  that  every  even  number  may  be  written  in 
the  form  2n  and  every  odd  number  in  the  form  2n-\-l 
where  ?^  represents  any  integer. 


MISCELLANEOUS   EXERCISE  40 

21.  Show  that  the  product  of  two  consecutive  numbers 
must  be  even  and  the  sum  o(hl. 

22.  Show  that  all  numbers  under  and  including  15  are 
factors  of  360360. 

23.  Find,  without  dividing,  the  remainder  after  364257 
lias  been  divided  by  3  ;   by  9  ;  by  11. 

24.  Evaluate  10  +  144  x  25  -  2180  -  15  +  5  x  3. 

25.  Write  4  numbers  of  5  places  each  that  are  divisible 
by  both  9  and  11. 

26.  Write  4  numbers  of  6  places  each  that  are  divisil)le 
by  both  3  and  6. 

27.  Write  4  numbers  of  4  places  each  that  are  divisible 
by  4,  5,  6. 

28.  Evaluate  47  x  68  +  68  x  53. 

29.  Evaluate  346  x  396.84  -  146  x  396.84. 

30.  Evaluate  27x3.1416-41x3.1416  +  49x3.1416 
^Qb  X  3.1416. 

31.  If  lemons  are  20  ct.  a  dozen  and  oranges  are  25  ct., 
how  many  oranges  are  worth  as  much  as  12|  doz.  lemons  ? 

32.  A  farmer  received  6  lb.  of  coffee  in  exchange  for 
9  doz.  eggs  at  12|  ct.  a  dozen.  How  much  was  the  coffee 
worth  per  pound  ? 

33.  Two  piles  of  the  same  kind  of  shot  weigh  respectively 
1081  lb.  and  598  lb.  What  is  the  greatest  possible  weight 
of  each  shot '/ 


FRACTIONS 

66.  Historically  the  fraction  is  very  old.  A  manuscript  on  arithme- 
tic, entitled  "  Directions  for  obtaining  a  Knowledge  of  All  Dark  Things," 
written  by  Ahmes,  an  Egyptian  priest,  about  1700  B.C.,  begins  with 
fractions.  In  this  manuscript  all  fractions  are  reduced  to  fractions 
with  unity  as  the  numerator.     Thus,  the  first  exercise  is  f  =  ^  +  ^^. 

67.  AVhile  the  Egyptians  reduced  all  fractions  to  those  with  con- 
stant numerators,  the  Babylonians  used  them  with  a  constant  de- 
nominator of  60.  Only  the  numerator  was  written,  with  a  special 
mark  to  denote  the  denominator.  This  method  of  writing  fractions 
lacked  only  the  symbol  for  zero  and  the  substitution  of  the  base  10 
for  60  to  become  the  modern  decimal  fraction.  Sexagesimal  frac- 
tions are  still  used  in  the  measurement  of  angles  and  time. 

68.  The  Romans  used  duodecimal  fractions  exclusively.  They 
had  special  names  and  symbols  for  J^,  j\,  •  •  •,  ii,  ^\,  J^,  etc.  To 
the  Romans,  fractions  were  concrete  things.  They  never  advanced 
beyond  expressing  them  in  terms  of  uncia  (J^),  silicus  (i  uncia), 
scrupulum  (^^  uncia),  etc.,  all  subdivisions  of  the  as,  a  copper  coin 
weighing  one  pound. 

69.  The  sexagesiinal  and  duodecimal  fractions  prepared  the  way  for 
the  decimal  fraction,  which  appeared  in  the  latter  part  of  the  sixteenth 
century.  In  1.58.5  Simon  Stevin  of  Bruges  published  a  work  in  which 
he  used  the  notations  7  4'  6"  5'"  9"",  or  7®  4®  6(2)  5®  00  for  7.4659. 
During  the  early  stage  of  its  development  the  decimal  fraction  was 
written  in  various  other  forms,  among  which  are  found  the  foUow- 

I    II  III  IV  12    3    4 

ing:  7  4  6  5  9,  7  4  6  5  9,  7 1 4659,  7  [4659 ,  7,4659.  The  decimal 
point  was  first  used,  in  1612,  by  Pitiscus  in  his  trigonometrical  tables ; 
but  the  decimal  fraction  was  not  generally  used  before  the  beginning 
of  the  eighteenth  century. 

40 


FR  ACT  IONS  47 

70.  Tlie  primary  conception  of  a  fraction  is  one  or  several 
of  tJie  equal  parts  of  a  unit.  i'lius,  the  fraction  |  indicates 
that  4  of  tlie  5  e(|nal  parts  of  a  nnit  are  taken. 

71.  The  term  namin;/  the  number  of  parts  into  which 
tlie  unit  is  divided  is  called  tlie  denominator.  The  term 
numbering  the  parts  is  called  the  numerator. 

72.  A  proper  fraction  is  less  than  unity  ;  an  improper 
fraction  is  ecjual  to  or  greater  than  unity. 

73.  A  number  consisting  of  an  integer  and  a  fraction  is 
called  a  mixed  number. 

74.  Our  conception  of  a  fraction  must,  however,  be 
enlarged  as  we  proceed,  and  be  made  to  include  such  ex- 

2  5       3  3 

pressions    as  -~^^  — -^    3.14159  •  •  -,  1,   etc.     The   more 
o.2o    —  i  ^ 

general  conception  of  a  fraction  is  that  it  is  an  indicated 

operation  in  division  where  the  numerator  represents  the 

dividend  and  the  denominator  the  divisor. 

75.  Decimal  fractions  are  included  in  the  above  defini- 
tion, as  0.5  means  ^-^  or  J.  Using  the  decimal  point  (0.5) 
is  simply  another  way  of  writing  y^Q^. 

76.  General  Principles.  3IuItipJ//ing  the  ^mmerator  or 
dividing  the  denominator  of  a  fraction  by  a  number  midti- 
plies  the  fraction  by  that  number. 

Let  Y  be  any  fraction  where  7  represents  the  number  of  parts  into 
which  unity  is  divided,  and  5  the  number  of  these  parts  taken. 

(1)  '  =  3  X  I;,  since  there  are  three  times  as   many  of  the  7 

7  7 

parts  of  unity  as  before. 


48  FRACTIONS 

5  5     .  .       . 

(2)  =  3  X  -.  since  dividing  the  denominator  by  3  divides  by 

7^37 

3  the  number  of  equal  parts  into  which  unity  is  divided  and  there- 
fore the  fraction  is  3  times  as  large  as  before. 

77.  Dividing  the  numerator  or  multipli/in;i  the  denomina- 
tor of  a  fraction  hy  a  number  divides  the  fraction  by  that 
number. 

(1)  — — -  =  -  -^  4,  since  dividing  the  numerator  by  4  divides  by  4 
the  number  of  parts  taken  without  changing  the  value  of  the  parts. 

(2)  =  -  -=-  4,  since  multiplying  the  denominator  b}*  4  multi- 

y  X  4     y 

plies  by  4  the  number  of  parts  into  which  unity  is  divided  and  there- 
fore the  fraction  is  \  as  large  as  before. 

78.  Multiplyi7ig  or  dividing  both  numerator  and  de- 
nominator of  a  fraction  by  the  same  number  does  not  cha)ige 
the  value  of  the  fraction. 

5  X  v*     o 

(1)  '- =  -•     Multii^lying  both  numerator  and  denominator  by 

5  X  3      3 

5  both  multiplies  and  divides  the  value  of  the  fraction  b}^  .5.     The 

value  of  the  fraction  therefore  remains  unchanged. 

9^5       9 
C2)  "  =  -•     Dividing  both  numerator  and  denominator  bv  5 

^   ^  3  -  .5      3  ^ 

both  divides  and  multiplies  the  value  of  the  fraction  by  5.     The  value 

of  the  fraction  therefore  remains  unchanoed. 


79.  A  mixed  number  may  be  reduced  to  an  improper  frac- 
tion and  an  improper  fraction  may  be  reduced  to  a  mixed 
number  or  an  integer. 

Thus,  5|  =  '--^ — ^—-.     Since  5x4  =  the  number  of  4ths  in  5  and 
4 

5x4  +  3  =  the  number  of  4ths  in  5 J,  .-.  o^  =  ^^. 

Reversing  the  process, 


V  =  23  -  4  =  5  +  I  =  5f. 


FRACTIONS  49 

80.  When  the  numerator  and  dejiominator  of  a  fraction 
are  prime  to  each  other^  the  fraction  is  said  to  he  in  its  low- 
est terms. 

Ex.    Express  ||  in  its  lowest  terms. 

42  ^  3  X  14  ^  3 
70      5  X  14     5* 

81.  Two  or  more  fractions  may  he  reduced  to  equivalent 
fractions  having  a  common  deno7ninator . 

Ex.  1.  Reduce  |,  f ,  ^^2'  ^^  equivalent  fractions  having  a 
common  denominator. 

The  1.  c.  m.  of  4,  9,  12,  is  36. 


3 

3x9 

27 

4 

4x9 

36 

5 

5x4 

20 

9 

9x4 

36 

1  _ 
12 

1  X  3 
12  X  3 

_  3 
36 

•*•  §6?  ih  3%  ^^'^  fractions  having  a  common 
denominator,  equivalent  to  |,  f,  j\.  Since  36  is 
the  1.  c.  ni.  of  4,  9,  12,  it  is  called  the  least  common 
denominator. 


82.  Sometimes,  instead  of  finding  the  l.c.m.,  it  is  more 
convenient  to  take  as  the  common  denominator  the  product 
of  all  the  denominators  and  multiply  each  numerator  by 
the  product  of  all  the  denominators  except  its  own. 

Ex.  2.  Reduce  |,  ^  and  |  to  fractions  having  a  common 
denominator. 

3  3x6x3      54  Since  the  common  denominator  is  4  x  6  x  3, 

4  ~  4  X  6  X  3  ~  72  4  is  contained  in  it  6  x  3  times  and  the  first 
numerator  will  be  3  x  6  x  3,  6  is  contained  in 
the  common  denominator  4x3  times  and  the 
second  numerator  will  be  1  x  4  x  3,  3  is  contained 


1      1x4x3      12 


6      4x6x3      7 

2  _  2x4x6  _  48       in  the  common  denominator  4x6  times  and  the 

3  4x6x3      72       third  numerator  will  be  2  x  4  x  6. 

lyman's  adv.  ar. — 4 


50  FB  ACTIONS 

83.  Addition  and  Subtraction  of  Fractions.  Since  only 
the  same  kinds  of  units,  or  the  same  parts  of  units,  can  be 
added  to  or  subtracted  from  one  anotlier,  it  is  necessary 
to  reduce  fractions  to  a  common  denominator  before  per- 
forming the  operations  of  addition  or  subtraction. 

Ux.  1.  Add  y%,  3^g  and  -^j. 

The  1.  c.  m.  of  12,  36  and  84  is  252. 

5  ^  5  X  21  ^  105     7  ^  7x7  ^  49      1^1x3^    3 
12      12  X  21      252'    36      36  x  7      252'    84     84  x  3     252' 

5       7        1  ^  105      49         3    ^  157 
12      36      84      252      252      252      2.52' 

Ux.  2.   Add  2|,  If  and  3lf . 

2|  +  If  +  3H  -  2  +  1  +  3  +  I  +  f  +  If  =  6  +  }- ft  +  j\%  +  j\%  =  7t\V 

After  a  little  practice  the  student  should  be  able  to 
abbreviate  the  work  very  much.     &.  1  might  be  worked 

briefly,  thus  ; 

5       7        1  ^  105  +  49  +  3  ^  157 
12      36      84  252  252'      . 

Ux.  2,  thus : 

9.  +  u  ,  313  _  6  +  l_0_8  +  80  +  39  _  . 

Ex.  3.   From  22^^^  subtract  181|. 


21f  I  -  18H  =:  3||  =  3if 

84.  Multiplication  of  Fractions.  The  product  of  two 
numbers  may  he  found  hy  performing  the  same  operation  on 
one  of  them  as  is  performed  on  unity  to  produce  the  other. 

Thus,  in  3x4  =  12,  nnity  is  taken  three  times  to  produce  the  mul- 
tiplier 3,  hence  4  is  taken  three  times  to  produce  the  product  12. 
Again,  in  |  x  ^  =  ^f ,  unity  is  divided  into  3  parts  and  2  of  them  are 


FRACTIONS  51 


taken  to  produce  the  multiplier  |,  hence,  5  is  divided  into  .3  parts,  each 

h  is   — '- — 
3x7 

2  X  5      1.0 


of  which  is   ——-   (Why?),  and  2  of  them  are  taken  to  produce  the 
8  X  /  ^ 


in-oduct 

^  3  X  7      21 

When  the  multiplier  is  a  common  fraction,  the  sign  (  x  )  should  be 

read  " of."     Thus,  |  x  $5  means  |  of  1 5. 

Ex.  1.    Multiply  II  by  fJ. 

1  1 

23  y^\  ^n  x  2X^1 
42      69      12  X  ^p      e' 

2  3 

85.  The  student  should  use  cancellation  luhenever  jjossible. 
He  shoidd  never  midtiply  or  divide  until  all  possible  factors 
have  been  removed  by  cancellation. 

Note.  Although  a  knowledge  of  the  principles  of  multiplication 
and  division  of  decimals  has  been  assumed  in  examples  given  before, 
it  is  well  to  review  these  principles  at  this  point  to  make  sure  that 
they  are  thoroughly  understood. 

Ex.  2.    Multiply  0.234  by  0.16. 

Solution.     0.234  x  0.16  =  '^  x  ^  0.234 

1000      100  Q  ^g 


234  X  16 
1000  X  100 


234 
1404 


•^'^^^    =0.03744.  0<03744 


100000 


The  number  of  decimal  places  in  the  product  is  the  same  as  the 
number  of  zeros  in  the  denominator  of  the  product,  that  is,  it  equals 
the  number  of  decimal  places  in  the  multiplicand  plus  the  number  in 
the  multiplier.  The  decimal  point  in  (0.03744)  simply  provides  a 
convenient  way  of  writing  xtwjo-  1^  ^^  better,  however,  to  determine 
the  position  of  the  decimal  point  before  beginning  the  multiplication. 
This  can  be  done  by  considei-ing  only  the  last  figure  at  tlie  right  of 
the  multiplier  and  multiplicand.  Thus,  we  see  that  0.004  x  0.06  = 
0.00024.     Hence,  the  order  of  the  product  will  be  hundred  thousandths. 


52  FRACTIONS 

86.  Tlie  following  simple  truths,  or  axioms,  are  fre- 
quently used  in  arithmetic. 

(1)  Numbers  that  are  equal  to  the  same  number  are  equal 
to  each  other. 

Thus,  if  X  =  5  and  //  =  5,  then  x  =  y. 

(2)  If  equals  are  added  to  equals.,  the  sums  are  equal. 
Thus,  if  X  =  5,  X  +  3  =  5  +  3. 

(3)  If  equals  are  sid^tracted  from  equals.,  the  remainders 
are  equal. 

Thus,  if  a;  =  4,  then  a;  -  2  =  4  -  2. 

(4)  If  equals  are  midtlplied  by  equals.,  the  products  are 
equal. 

Thus,  if  ^  =  3,  then  x  =  Q. 

(5)  //'  equals  are  divided  by  equals^  the  quotients  are 
equal. 

Thus,  if  3  X  =  G,  tlien  x  =  2. 

87.  Division  of  Fractions.  Division  may  be  regarded  as 
the  inverse  of  multiplication.  The  problem  is,  therefore, 
to  find  one  of  two  factors  when  the  product  and  the  other 
factor  are  given. 

Thus,  3x4=  12,  .-.  12  -3  =  4,  and  12  -  4  =  3.        Axiom  5. 


To  divide  one  fraction  by  another. 

Solution.     Let  ||  -^  |  =  7  (a  quotient). 

Then  ||  =  |  x  r/  (multiplying  both  members  of  the  equation  by  f), 
and  II  X  f  =  7  (multiplying  both  members  of  the  equation  by  f). 

••.  the  quotient  is  obtained  by  nmltiphjing  the  dividend  by  the  reciprocal, 
of  the  divisor. 


FRACTIONS  53 

^a?.    Find  the  value  of  .]  +  2  x  |  x  J  +  [;  ^  :^  x  2  -  J. 


2      5      3      4      (3      ii      7      5      2      5      )J      0      7      5 
o 

Observe  tliat  in  the  above  exercise  the  fractions  connected  by  x 
or  -f-  are  first  operated  upon,  then  those  connected  by  +  or  — . 

3 
88.    A  fraetion  of  the  form  |-  is  called  a  complex  frac- 

6 

tion  and  may  be  considered  as  equivalent  to  |  -J- 1  and 
treated  as  a  problem  in  division.  In  general,  however,  a 
complex  fraction  may  be  more  readily  simplified  by  multi- 
plying both  terms  by  the  1.  c.  m.  of  the  denominators  of 
the  two  fractions  in  the  numerator  and  denominator, 

^^   1     f_63xf_35 

Rv.  2.    Divide  38.272  by  7.36.  5.2 

7.36)38.272 

Solution.     38.272  -  7.36  =  \%'J^^  -  lU  36.80 

—  SAT?  2.  y    log  _   38212    y      100_  —   5^   x   J-  —  ,0  '>  ,     ,„^ 

—  1000     ^^73  6^—      736      -^TOOO"     1     -^10—  ^— '  1.472 

1.472 

The  number  of  decimal  places  in  the  quotient  will  equal  the  num- 
ber of  zeros  in  tiie  denominator  of  the  last  product.  This  will  be  the 
same  as  the  number  of  zeros  in  the  denominator  of  the  dividend  minus 
the  number  of  zeros  in  the  denominator  of  the  divisor,  or,  Avhat  is  the 
same  thing,  the  number  of  decimal  places  in  the  dividend  minus  the 
number  of  decimal  places  in  the  divisor. 

If  the  number  of  decimal  places  in  the  dividend  is  less  than  the 
number  of  decimal  places  in  the  divisor,  we  may  annex  zeros  to  the 
dividend  till  the  number  of  decimal  places  is  the  same  in  both  divi- 
dend and  divisor.     The  quotient  up  to  this  point  in  the  division  will 


54  FB  ACTIONS 

be  an  integer,  and,  in  case  it  is  necessary  to  carry  the  division  farther, 
more  zeros  may  be  annexed  to  the  dividend.  The  remaining  figures 
of  the  quotient  will  be  decimals. 

Ux.  3.    Divide  52.36  by  3.764. 

13.9 

3.764)52.360|0 
37.64 


14.720 
11.292 
3.4280 
3.3876 
404 


EXERCISE   16 

1.  Change  |  to  9tbs ;   2^  to  168ths. 

2.  Reduce  to  lowest  terms  each  of  the  following  frac- 

tinim  •    -9-     111     -7-2-      1128 

3.  Explain  the  reduction  of  7|  to  an  improper  fraction. 

4.  Explain  the  reduction  of  -^^5^-  to  a  mixed  number. 

^.      ,.,     4f    I    0.5    0.75 

5.  Simplify  ^,  |,  -J-,  -^. 

^6323 

fi       Arlrl     3668         1221      r,nfl         5 
b.     i\aa   10  9  8  9'     13  4  31    ^^^^^     12  2  1- 

7.  From  75^^2  ^-^ke  12-\. 

8.  Multiply  21  -  f  by  J  of  f  X  |. 

9.  Find  the  value  of  f  of  ^3_  _^  2  ><  _6_  of  |f . 

10.  Find  the  value  of  1-^f  of  f  x|-|-^|  of -|--fo-^f 

11.  Find  the  value  of  5|-0.9  of  2.7  +  251x0.02-^. 

12.  rind  the  value  or  — i-^ 


4is  *'t  -^T 


•7    1 


FRACTIONS  55 

13.  By  what  must  \  he  multiplied  to  produce  -SJ  ? 

14.  Wliat  number  divided  by  -V^^-  of  |  will  give  4|  as  a 
quotient  ? 

Simi)lifv:    ^-l-i:3j^.G ^>ij7:. 


15.   Simplify:    ^-^^^-^^ ^^- 

1   _  1    of   1  ^  4-  3   V   4 
-■■        5  ^^   13  +  ?  >^  Y 


16.  What  fraction  added  to  the  sum  of  |,  ^,  and  5.25 
will  make  6.42  ? 

17.  Simplify:    /  ^^//^ ^^  • 

^      "^      5  +  0.5  of  (Jg-0.9) 

18.  How  is  the  value  of  a  proper  fraction  affected  by 
adding  the  same  number  to  both  numerator  and  denomi- 
nator ?    How  is  the  value  of  an  improper  fraction  affected  ? 

19.  A  merchant  bought  a  stock  of  goods  for  f  2475.50 
and  sold  -|-  of  it  at  an  advance  of  \  of  the  cost,  \  of  it  at  an 
advance  of  ^  of  the  cost,  and  the  remainder  at  a  loss  of  ^^ 
of  the  cost.     Did  he  gain  or  lose  and  how  much  ? 

20.  A  ship  is  worth  $90,000  and  a  person  who  owns  -f^^ 
of  it  sells  \  of  his  share.  What  is  the  value  of  the  part 
he  has  left  ? 

21.  If  1  is  added  to  both  numerator  and  denominator  of 
|,  by  how  much  is  its  value  diminished  ? 

22.  If  1  is  added  to  both  numerator  and  denominator 
of  |,  by  how  much  is  its  value  increased  ? 

89.  Cancellation.  Much  time  may  be  saved  in  solving 
problems  by  w^riting  down  a  complete  statement  of  the 
condition  given  and  then  canceling  common  factors  if  any 
are  present.  The  student  should  do  this  at  every  stage  in 
the  solution  of  a  problem,  always  factoring  and  canceling 
whenever  possible,  and  never  multiplying  or  dividing  till  all 
2?osi<ihle  factors  have  been  removed  by  cancellation. 


56  FRACTIONS 

Ex.  1.  If  -^-^  of  a  business  block  is  worth  $6252.66, 
what  is  the  value  of  ^  of  it  ? 

Solulion.     £j  is  worth  ^  6252.66. 

2V  is  worth  1  of  ^  62.52.66. 

.-.  the  whole  is  worth  ^  of  $  62.52.66. 

.-.  {i  is  worth  i^  of  V  of  $  6252.66. 

347.37 

5      xm-u 

=  11  X  2^  X  $0232.06  ^  ^  ^9^0-35^ 

^a;.  2.  How  much  must  be  paid  for  59,400  lb.  of  coal  at 
$  4  per  ton  of  2000  lb.  ? 

Statement.     59400  x  $4  ^  ^  ^^3 
2000 

EXERCISE   17 

Find  the  value  of : 

27  X  72  X  80  1320  x  432  x  660 

*    86x45x30*  *   4400x297x288* 

2    144  X  1728  X  999  ^    1760x9x125 

96  X  270  X  33    *  '        55  x  360 

.   5.    How  much  must  be  paid  for  shipping  1200  bbl.  of 
apples  at  i  35  per  hundred  barrels  ? 

6.  How  many  bushels  of  potatoes  at  50  ct.  a  bushel 
will  pay  for  500  lb.  of  sugar  at  4  ct.  a  pound  ? 

7.  A  merchant  bought  12  carloads  of  apples  of  212  bbl. 
each,  3  bu.  in  each  barrel  at  45  ct.  per  bushel.  He  paid 
for  them  in  cloth  at  25  ct.  per  yard.  How  many  bales 
of  500  yd.  did  he  deliver? 

8.  How  many  bushels  of  potatoes  at  55  ct.  per  bushel 
must  be  given  in  exchange  for  22  sacks  of  corn,  each  con- 
taining 2  bu.,  at  60  ct.  a  bushel? 


APPROXIMATE  RESULTS  57 

APPROXIMATE   RESULTS 

90.  In  scientific  investigations  exact  results  are  rarely 
possible,  since  the  numbers  used  are  obtained  by  observa- 
tion or  by  experiments  in  wliit'li,  however  fine  the  instru- 
ment, the  results  are  only  approximate,  and  there  is  a 
degree  of  accuracy  beyond  which  it  is  impossible  to  go. 

91.  On  the  other  hand,  the  approximate  value  of  such 
incommensurable  quantities  as  V2  =  1.414 +  ,  tt  =  3.14159  + 
can  be  obtained  to  any  required  degree  of  accuracy.  The 
value  of  TT  has  been  computed  to  707  decimal  places,  but 
no  such  accuracy  is  necessary  or  desirable.  The  student 
should  always  bear  in  mind  that  it  is  a  tvaste  of  time  to 
carry  out  results  to  a  greater  degree  of  accuracg  than  the 
data  on  ivhich  they  are  founded. 

92.  It  is  frequently  necessary  to  determine  the  value  of 
a  decimal  fraction  correct  to  a  definite  number  of  decimal 
places.  The  value  of  ||  =  0.95833+  correct  to  four  deci- 
mal places  is  0.9583.  0.958  and  0.96  are  the  values  correct 
to  three  and  two  places.  The  real  value  of  this  fraction 
correct  to  four  places  lies  between  0.9583  and  0.9584. 
0.9583  is  0.00003+  less  than  the  true  value,  while  0.9584 
is  0.00006+  greater.  Therefore  0.9583  is  nearer  the  cor- 
rect value,  and  is  said  to  be  the.  value  correct  to  four  deci- 
mal places.  Similarly,  0.96  is  the  value  correct  to  two 
places. 

93.  If  5  is  the  first  rejected  digit,  the  result  will  appar- 
ently be  equally  correct  whether  the  last  digit  is  increased 
by  unity  or  left  unchanged.  The  value  of  0.4235  correct 
to  three  places  may  be  either  0.423  or  0.424.  However, 
as  the  5  itself  is  usually  an  approximation,  it  can  readily 


58  FRACTIONS 

be  determined  which  course  to  pursue  by  noticing  whether 
the  5  is  in  excess  or  defect  of  the  correct  value.  0.23649 
correct  to  four  places  is  0.2365,  but  correct  to  three  places 
0.236  is  nearer  the  true  value  than  0.237. 

94.  Addition.     Ux.    Add  0.234678,  0.322135,  0.114342, 

0.568217,  each  fraction  being  correct  to  six  decimal  places. 

Solution.     It  is  clear  that  the  last  digit  in  this  sum  is 

not  correct,  since  each  of  the  four  numbers  added  may  be  0.-34:673 

either  greater  or  less  than  the  correct  value  by  a  fraction  0.322135 

less  than  0.0000005.     Hence,  the  total  error  in  the  sum  0.114342 

cannot  be  greater  or  less  than  0.000002.      The  required  0.563217 

sum  must  therefore  lie  between  1.234369  and  1.234365,  and  1.234367 
in  either  case  the  result  correct  to  five  places  is  1.23437. 

95.  The  next  to  the  last  digit  in  the  sum  may  be  incor- 
rect, as  shown  in  the  following  example : 

Ux.    Add  0.131242,  0.276171,  0.113225,  0.342247,  each 

fraction  being  correct  to  six  decimal  places. 

0.131242 

Solution.     In  this  case  the  sum  lies  between  0.862887      0.276171 

and  0.862883.     Hence,  it  is  uncertain  whether  0.86289  or      0.113225 

0.86288  is  the  value  correct  to  five  places.     0.8629  is,  how-      0.342247 

ever,  the  value  correct  to  four  places.  • 

^  0.862885 

96.  The  third  digit  from  the  last  may  be  left  in  doubt, 
as  in  the  following  example : 

Ux.    Add  5.866314,  3.715918,  0.568286,  4.342238,  each 

fraction  being  correct  to  six  places. 

5.866314 

Solution.     Here  the  true  value  of  the  sum  lies  between      3.715PI8 

14.492753  and  14.492749.     Hence,  it  is  uncertain  whether      0.568286 

the  value  correct  to  four  places  is  14.4928  or  14.4927.    The      4.842233 

value  correct  to  three  places  is  14.493.  TTTTT^^TT 

14.492 /ol 


APPROXIMATE  RESULTS  59 

97.  Subtraction.  Ux.  Subtract  0.288047  from  0.329528, 
each  fraction  hciug  correct  to  six  decimal  places. 

Solution.  Since  each  fraction  cannot  differ  from  the  O.;i20o28 
true  value  by  a  fraction  as  large  as  0.0()00()(>5,  the  differ-  0.2  )."S<;47 
ence  cannot  be  greater  or  less  than  the  correct  value  by  a  0.0!M)yi5l 
fraction  as  large  as  0.000001.  Hence,  the  difference  must 
lie  between  0.090882  and  0.090880,  and  the  value  correct  to  five  places 
is  0.09088. 

98.  Cases  will  arise  where  the  second  and  third  digits 
from  the  last  are  in  doubt,  as  in  addition.  The  student 
should  determine  how  far  the  result  may  be  relied  upon 
in  the  following  examples  : 

(1)  Subtract  0.371492  from  0.764237. 

(2)  Subtract  0.11132  from  0.23597. 

(3)  Subtract  15.93133  from  43.71288. 

99.  Multiplication.  From  the  examples  in  addition 
given  above  the  student  will  notice  that  it  will  be  suffi- 
cient in  most  cases  to  carry  out  the  partial  products 
correct  to  two  places  more  than  the  required  result. 

Ex.  Find  the  square  of  3.14159  correct  to  four  decimal 
places. 

Solution.  The  multiplication  in  full  and  the  contracted  form  are 
as  follows:         3^^^.^  3^^^.^ 

3.14159                          •  3.14159 

9.42477  9.42477 

.3141.59  .314159 

.1256636  .125664 

.00314159  3142 

.001570795  1571 

.0002827431  283 

9.8695877281  9.8696 

After  pointing  off  the  first  partial  product  we  proceed  as  indicated 
in  the  above  contracted  form  until  the  multiplication  by  3  and  1  are 


60  FRACTIONS 

completed.  Multiplication  by  4  would  give  a  figure  in  the  seventh 
place.  •  Instead  of  writing  down  the  figures  we  add  the  nearest  10  to 
the  next  column.  Thus,  4  times  9,  36,  add  4  to  the  next  column  since 
3.6  =  4  apj^roximately.  4  times  5,  20  and  4,  24.  4  times  1,  4  and  2, 
6,  etc. 

In  multiplying  by  the  next  1  it  is  not  necessary  to  take  the  9  in  the 
multiplicand  into  account.  So,  also,  in  multiplying  by  the  5,  the  5 
and  9  in  the  multiplicand  may  both  be  ignored.  And  so  on  until  the 
multiplication  is  completed. 

100.  Division.  JSx.  Divide  9376245  by  3724  correct  to 
the  units'  place. 

Solution.  The  division  in  full  and  the  contracted  form  are  as 
follows  : 


2517 
19376245 

7448 

2517 
3724)9376245 
7448 

19282 
18620 

19282 
18620 

6624 
3724 

662 
372 

29005 
26068 

290 
260 

2937 

30 

The  first  two  digits  in  the  quotient  are  2  and  5  and  the  second  re- 
mainder is  662.  It  is  not  necessary  to  bring  down  any  more  figures 
to  have  a  result  correct  to  units  since  tens  divided  by  thousands  will 
give  hundredths.  The  divisor  may  also  be  contracted  at  this  stage  of 
the  work.  Thus,  cutting  off  the  4,  372  is  contained  once  in  the  second 
remainder,  662.  Cutting  off  the  2,  37  is  contained  7  times  in  the  next 
remainder,  290.     This  gives  the  units'  figure  of  the  quotient. 

2517 

It  will  be  noticed  that  the  next  figure  of  the  quo-     3704.^9376045 

tient  is  greater  than  0.5,  therefore  the  result  correct  19'>8"^ 

to  units  is  2518.  ggo 

The  work  may  be  further  abridged  by  omitting  the  Qon 

partial  products  and  writing  down  the  remainders  only.  oq 


APPROXIMATE   RESULTS  61 

101.    Ex.    Divide  62.473  by  411).GT89.*  miQc-nn 

•^  O.1488o90 

Solution.     First  shift  the  decimal  point  410G7fSf>)0LU7;)O.<) 

four  places  in  each  so  as  to  have  an  integral  "JO.lOol  10 

divisor,  and  then  work  as  follows  :    'I'he  1  ^37 17!*")!: 

and  4  are  obtained  without  abbreviating  ^OO.l^;} 

and  the  8,  8,  5,  9,  0  by  cutting  off  9,  8,  7,  24780 

6,  9  in  succession  from  the  divisor.  3796 

19 

E.V.     Divide  0.0167  by  423.74.* 

0.00003941  *  From  Langley's  "Treatise 

42374)1.67000  on  Computation,"  p.  68. 

39878 

1741 

46 

4 

EXERCISE   18 

1.  Divide  100  by  3.14159  correct  to  0.01. 

2.  Find  the  quotient  of  67459633  divided  by  4327  cor- 
rect to  five  significant  figures. 

3.  Determine  witliout  dividing  by  what  number  less 
than  13,  339295680  is  exactly  divisible. 

Determine  by  casting  out  tbe  9\s  whether  the  following 
are  correct : 

4.  959x959  =  919681.   5.  954x954x954  =  868250664. 
6.   33920568-729  =  42829,    7.  1019x1019  =  1036324. 

8.  6234751  -  43265  =  14.41+  a  remainder  2645. 

9.  Find  the    sum  of    23.45617,   937.34212,  42.31759, 
532.23346,  141.423798  correct  to  two  decimal  places. 

10.  Subtract  987.642  from  993.624  correct  to  tenths. 

11.  Find  the  product  of  32.4736  x  24.7955  correct  to 
five  significant  figures. 

12.  Divide  47632  by  ^.       13.    :\Iultiply  23793  by  124- 


MEASURES 

102.  Measures  of  Weight.  It  is  curious  to  note  what  ati  important 
part  the  grain  of  wheat  or  barley  has  played  in  the  establishment  of 
a  unit  of  weight,  both  among  the  ancients  and  the  more  modern 
Europeans.  In  England,  as  early  as  1266,  we  find  the  pennyweight 
defined  as  the  weight  of  "  32  wheat  corns  in  the  midst  of  the  ear  " ; 
again  about  1600,  as  "  24  barley  corns,  dry  and  taken  out  of  the  middle 
of  the  ear."  Still  later  the  artificial  grain  (^\f  oz.  Troy)  is  defined  as 
"one  grain  and  a  half  of  round  dry  wheat."  The  Greeks  made  four 
grains  of  barley  equivalent  to  the  keration  or  carob  seed.  From  this 
is  derived  the  carat,  the  measure  by  which  diamonds  and  pearls  are 
weighed.  The  grain  of  barley  and  the  carat  have  been  used  by  all 
European  countries  as  the  basis  of  existing  weights. 

103.  Great  inconvenience  was  long  experienced  from  this  lack  of 
uniformity,  so  that  Parliament  in  1821  passed  an  act  adopting  the 
Imperial  Pound  Troy  as  the  standard  of  weight.  It  was  also  enacted 
that  of  the  5760  grains  contained  in  the  pound  Troy,  the  pound  avoir- 
dupois should  contain  7000.  The  unit  pound  is  defined  by  a  piece  of 
metal  kept  in  the  standard  office.  The  ounce,  grain,  etc.,  are  subdivi- 
sions of  the  pound. 

104.   Avoirdupois  Weight 

16  drams  (dr.)  =  1  ounce  (oz.) 
16  ounces  =  1  pound  (lb.) 

100  pounds  =  1  hundredweight  (cwt.) 

2000  pounds         :=  1  ton  (T.) 

112  lb.  =  1  long  cwt.  and  2240  lb.  =  1  long  ton  are  used  in  the 
customhouse  and  in  weighing  coal  and  iron  at  the  mines. 
The  c  in  cwt.  stands  for  the  Latin  word  centum,  a  hundred. 
Lb.  is  a  contraction  of  the  Latin  word  libra,  pound. 
Pound  is  from  the  Latin  word  pondus,  a  weight. 
Ounce  is  from  the  Latin  word  uncin,  a  twelfth  part. 
Dram  is  from  the  Latin  word  drachma,  a  handful. 

62 


ME As  CUES  G3 

105.    Tkoy  Wkkjiit 

24  grains  (gr.)      =  1  peiinyweiglit  (pvvt.) 

20  pennyweights  =  1  ounce  Troy 

12  ounces  Troy     =  1  ])oun(l  Troy 

This  weight  is  used  for  the  precious  metals  and  jewels.  The  ounce 
Troy  and  pound  Troy  must  be  carefully  distinguished  from  the  ounce 
and  pound  avoirdupois.     The  grain,  however,  is  the  same  throughout. 

437.5  grains  =  1  ounce  avoirdupois  480  grains  =  1  ounce  Troy 

7000  grains  =  1  pound  avoirdupois      5760  grains  =  1  pound  Troy 

106.    Apothecaries'  Weight 
20  grains     =  1  scruple  (sc.  or  9) 

3  scruples  =  1  dram  (di-.  or  3  ) 

8  drams     =  1  ounce  (oz.  or  ^  ) 
12  ounces    =  1  pound 
5700  grains     =  1  pound 

This  table  is  used  in  compounding  drugs  and  medicines.  Scruple 
is  from  the  Latin  word  scrupulum,  a  small  weight. 

Of  the  above  measures  of  weight,  avoh'dupois  is  the  most  generally 
used. 

107.  Measures  of  Length.  The  ancients  usually  derived  their  units 
of  length  from  some  part  of  the  human  body.  Thus,  we  find  the 
fathom  (the  distance  of  the  outstretched  hands),  the  cubit  (the  length 
of  the  forearm),  and  later  the  ell  (the  distance  from  the  elbow  to  the 
end  of  the  finger),  the  foot  (the  length  of  the  human  foot),  the  span 
(the  distance  between  the  ends  of  the  thumb  and  little  finger  when 
outstretched),  the  jJalm  (the  width  of  the  hand),  the  dir/if  (the  breadth 
of  the  finger).  The  Roman  foot  was  subdivided  into  four  palms, 
and  the  palm  into  four  digits.  The  division  into  inches  or  uncice  (a 
twelfth  part)  applied  not  only  to  the  foot  but  to  anything. 

108.  For  longer  measures  there  was  still  less  uniformity.  We 
find  the  Hebrew's  half-da fs  journey ;  the  Chinese  /t'A,  the  distance  a 
man's  voice  can  be  heard  upon  a  clear  plain  ;  the  Greek  stadium,  prob- 


64  MEASURES 

ably  derived  from  the  length  of  the  race  course ;  the  Roman  pace  of 
five  feet ;  the  furlong,  the  length  of  a  furrow.  The  mille  passus,  a 
thousand  paces,  is  the  origin  of  the  modern  7nile. 

109.  In  1374  the  inch  is  defined  in  English  law  as  the  length  of 
"  three  barley  corns,  round  and  dry."  Later,  other  arbitrary  measures 
of  length  were  adopted  by  the  government.  The  standard  unit  in 
England  and  the  United  States  is  the  yard.  The  standard  yard  is  the 
length  of  a  metal  bar  preserved  in  the  office  of  Standard  Weights  and 
Measures.  The  standard  foot  and  inch  are  subdivisions  of  this 
standard  yard. 

110.   Common  Measures  of  Length 

12  inches  (in.)  =  1  foot  (ft.) 

3  feet  =  1  yard  (yd.) 

5i  yards  or  16i  feet  =  1  rod  (rd.) 

320  rods  or  5280  feet  =  1  mile  (mi.) 

The  furlong,  equal  to  40  rods,  is  seldom  used. 

The  fathom,  equal  to  6  feet,  and  the  knot  or  geographical  mile, 
equal  to  one  minute  of  the  equatorial  circumference  of  the  earth 
(6080  feet),  are  sometimes  used. 

111.    Surveyors'  Measures  of  Length 
7.92  inches  =  1  link  (li.) 
100  links    =  1  chain  (ch.)  =  (4  rd.) 
80  chaifts  =  1  mile 

112.    ^Measures  of  Surfaces 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  square  feet  =:  1  square  yard  (sq.  yd.) 

30^  square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

1  square  mile  =  1  section. 

36  sections  =  1  township  (twp.) 


MEASURES  65 

113.   Measures  of  Solids 

1728  cubic  inclies  (cu.  iu.)  —  1  cubic  foot  (cu.  ft.) 

27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 

The  cord,  equal  to  128  cubic  feet,  is  a  rectangular  solid  8  feet  long, 
4  feet  wide,  and  -1  feet  high.  The  common  use  of  the  word  is,  how- 
ever, a  pile  of  wood  8  feet  long  and  4  feet  high,  the  widtli  of  the  pile 
varying  with  the  length  of  the  stick. 

1  cubic  yard  =  1  load 

24|  cubic  feet  =  1  perch 

114.  Measures  of  Money.  Originally,  among  primitive  people, 
buying  and  selling  was  carried  on  by  barter,  or  the  actual  exchange 
of  commodities.  The  inconveniences  arising  from  transactions  of 
this  kind  brought  about  the  adoption  of  a  medium  of  exchange,  or 
money.  Money,  usually  consisting  of  gold  and  silver,  was  used  at  a 
very  early  period  in  the  world's  history.  Gold  and  silver  seem  at  first 
to  have  been  exchanged  for  commodities  by  weight.  Business  trans- 
actions were  then  still  further  simplified  by  the  introduction  of  coins 
and  paper  money.  Finally,  as  in  the  case  of  weights  and  measures, 
governments  adopted  definite  standards  of  money  value. 

115.    United  States  Money 
10  mills     =  1  cent  (ct.) 
10  cents     =  1  dime  (d.) 
10  dimes   =  1  dollar  (%) 
10  dollars  =  1  eagle  (E.) 

116.  English  Money 

12  pence  (rl.)=  1  shilling  (.s.)  =  10.2433 
20  shillings    =  1  pound  (£)    =  ^4.8665 

117.  French  Money 
10  centimes  =  1  decline 

10  decimes  =  1  franc  =  .|  0.193 

LTMAX'S    ADV.  AR.  5 


66  MEASURES 

118.    German  Money 

100  pfennigs  =  1  mark  (M.)  =  80.238 

119.    jNIeasures  of  Number 

12  units  =  1  dozen  (doz.) 
12  dozen  =  1  gross  (gro.) 
12  gross  =  1  great  gross  (gt.  gro.) 
Also  24  sheets  of  paper  =  1  quire 
20  quires  =  1  ream 

120.    Liquid  Measure 

4  gills  (gi.)=l  pint  (pt.) 
2  pints  =  1  quart  (qt.) 

4  quarts       =  1  gallon  (gal.)  =  2-31  cu.  in. 
31i  gallons      =  1  barrel  (bbl.) 

121.    Dry  Measure 

2  pints    =  1  quart 

8  quarts  =  1  peck  (pk.) 

4  pecks   =  1  bushel  (bu.)=  21.10.42  cu.  in. 

The  Winchester  bushel  is  the  standard  nieusure  for  dry  substances. 
It  is  a  cylindrical  vessel  18|  in.  in  diameter  and  8  in.  deep,  containing 
2150.42  cu.  in. 

Before  the  adoption  of  this  and  other  standards  by  the  English 
government  there  was  even  a  greater  variety  of  measures  of  capacity 
than  of  length  and  weight. 

122.  Reduction  of  Compound  Numbers.  Quantities  like 
5  mi.  10  1(1.  7  yd.  2  ft.  iiiid  3  lb.  5  oz.  are  called  com- 
pound numbers,  because  they  are  expressed  in  several 
denominations. 


MEASURES  67 

Ux.  1.    Reduce  25  yd.  2  ft.  11  in.  to  inches. 


Solution.     1  yd.  =  3  ft. 

.-.  25  yd.  =  25  x  3  ft.  =  75  ft. 
75  ft.  +  2*^  ft.  =  77  ft.     1  ft.  =  12  in. 

.-.  77  ft.  =  77  X  12  in.  =  924  in. 
924  in. +11  in.  =  935  in. 


Note.  The  explanation  shows  that  25  and  77  are  the  multipliers  and  3  ft. 
and  12  in.  the  multiplicands;  but  to  shorten  the  operation,  3  and  12,  regarded 
as  abstract  numbers,  may  be  used  as  multipliers,  since  the  product  of  25  X  3  = 
the  product  of  3  x  25. 

Ex.  2.    Reduce  1436  pt.  to  bushels,  pecks,  etc. 

1436  no.  of  pt. 


25  or 

25  yd.         =  900  in. 

_3 

2  ft.         =  24  in. 

75 

11  in.         =  11  in. 

2  .-. 

77 

25  yd.  2  ft.  11  in.  =  935  in. 

12 
924 

11 

935 

718  no.  of  qt. 
89  no.  of  pk.  +  6  qt. 
22  no.  of  bu.  + 1  pk. 


Solution.  Since  there  are  2  pt.  in  1  qt., 
in  1436  pt.  there  are  as  many  quarts  as 
2  pt.  are  contained  times  in  1436  pt.,  or 
718  qt. 

Since  there  are  8  qt.  in  1  pk.,  in  718  qt. 
there  are  as  many  pecks  as  8  qt.  are  contained  times  in  718  qt.,  or 
89  pk.  6  qt. 

Since  there  are  4  pk.  in  1  bu.,  in  89  pk.  there  are  as  many  bushels 
as  4  pk.  are  contained  times  in  89  pk.,  or  22  bu.  1  pk. 

.-.  1436pt.=  22bu.  Ipk.  6qt. 

EXERCISE   19 

1.  Reduce  3  A.  5  sq.  rd.  12  sq.  yd.  to  square  yards. 

2.  Reduce  11000  sq.  rd.  to  acres. 

3.  Reduce  2  gt.  gro.  5  gro.  to  dozens. 

4.  Reduce  972  sheets  to  reams. 

5.  Reduce  20  cu.  yd.  to  cubic  inches. 

6.  Reduce  1000  oz.  to  pounds  and  ounces  (avoirdupois). 


68  MEASURES 

7.  Reduce  12  lb.  5  oz.  11  pwt.  20  gr.  to  grains. 

8.  Reduce  113  T.  7  cwt.  11  lb.  to  pounds. 

9.  Reduce  14763051  lb.  to  tons. 

10.  Reduce  5  sq.  yd.  3  sq.  ft.  91  sq.  in.  to  square  inches. 

11.  Reduce  46218385  sq.  in.  to  acres. 

12.  Reduce  5  bu.  7  pk.  3  qt.  to  Cjuarts. 

13.  Reduce  34372  pt.  to  pecks. 

14.  Reduce  21  yd.  to  a  decimal  of  a  mile. 

15.  Reduce  2  pk.  3  qt.  1  pt.  to  a  decimal  of  a  bushel. 

16.  Reduce  0.0125  A.  +  0.25  sq.  rd.  to  square  feet. 

17.  Reduce  0.01  of  a  cubic  yard  to  cubic  inches. 

18.  Reduce  43629145  in.  to  miles. 

19.  Reduce  |  of  a  peck  to  pints. 

20.  Reduce  2  qt.  1  pt.  to  a  fraction  of  a  peck. 

21.  Reduce  1  mi.  11  ch.  to  feet. 

123.    Addition   and  Subtraction   of   Compound  Numbers. 

Compound  addition  and  subtraction  is  tlie  addition  and 
subtraction  of  compound  numbers  of  the  same  kind.  The 
processes  differ  very  little  from  the  corresponding  pro- 
cesses in  the  addition  and  subtraction  of  abstract  numbers. 

Ex.  1.    Add  4  lb.  7  oz.  (Av.),  3  lb.  4  oz.,  12  11).  10  oz., 
9  lb.  5  oz. 

Solution.     The  work  is  as  follows : 
5,  1.5,  19,  26  oz.  =  1  lb.  10  oz. 
1,  10,  22,  25,  20  lb. 


4  lb 

/  oz 

3 

4 

12 

10 

9 

5 

291b. 

10  oz. 

MEASURES  69 

Ux.  2.    From  41  lb.  4  oz.  (Av.)  subtract  29  lb.  8  oz. 

The  work  is  as  follows  :  41  lb       4  oz 

1  lb.,  or  16  oz.  +  4  oz.  =  20  oz.  29  8 

8  and  12  are  20  TTVi      To 

,7:;:      .  ..        .-.  11  lb.    12 oz. 

1  and  29  and  11  are  41. 

124.  Multiplication  of  Compound  Numbers. 

Ux.  Multiply  5  yd.  2  ft.  by  7. 

7  X  2  ft.  =  14  ft.  =  4  yd.  2  ft.  5  yd.  2  ft. 

7x5  yd.  =  35  yd.  7 

35  yd.'+  4  yd.  2  ft.  =  39  yd.  2  ft.  39  yd.  2  ft. 

125.  Division  of  Compound  Numbers.  Compound  divi- 
sion is  of  two  kinds.  The  first  is  the  converse  of  multi- 
plication. In  this  case  the  quotient  is  a  compound 
number  of  the  same  kind  as  the  dividend.  In  the  second 
case  the  dividend  and  divisor  are  both  compound  numbers 
of  the  same  kind,  and  the  quotient  is  an  abstract  number. 

126.  The  two  cases  arise  from  the  fact  that  division  may 
be  regarded  as  the  operation  of  finding  one  of  two  factors 
when  the  other  factor  and  the  product  are  given. 

Thus,  39  yd.  2  ft.  is  the  product  of  7  and  5  yd.  2  ft. 

...  3^^^-^^^-  =  5  yd.  2  ft.,  or  '^ll^^AJ^  =  ^1^=7,  the  divi- 
7  ^  '  5  yd.  2  ft.        17  ft. 

dend  and  divisor  being  reduced  to  the  same  denomination  before 

dividing. 

Ux.  1.    Divide  29  mi.  2  yd.  2  ft.  by  8. 

Solution.     29  mi.  ^8  =  3  mi.  +  a  remainder  of  5  mi. 

5  mi.  =  8800  yd.  and  8800  yd.  +  2  yd.  =  8802  yd. 

8802  yd.  -f-  8  =  1100  yd.  +  a  remainder  of  2  yd. 

2  yd.  =  6  ft.,  and  6  ft.  +  2  ft.  =  8  ft. 

8  ft.  -  8  =  1  ft. 

.♦.  29  mi.  2  yd.  2  ft.  -  8  =  3  mi.  1100  yd.  1  ft. 


70  ME  A  SUB  ES 

Ux.  2.    Divide  139  lb.  8  oz.  (Av.)  by  4  lb.  8  oz. 

Solution.     139  lb.  8  oz.  =  2232  oz. 
4  lb.  8  oz.  =  72  oz. 
2232  0Z. -^72oz.  =  31. 

127.  Check.  Compound  addition  and  subtraction  may- 
be checked  in  the  same  way  as  addition  and  subtraction 
of  simple  numbers.  Multiplication  may  be  checked  by 
division,  and  division  by  multiplication. 


EXERCISE   20 

1.  How  many  inches  are  there  in  1  mi.  3  ch.  ? 

2.  Add  14  lb.  3  oz.,  5  lb.  7  oz.,  31  lb.  11  oz. 

3.  From  17  cu.  yd.  11  cu.  in.  subtract  5  cu.  yd.  5  cu.  ft. 

4.  From  11  bu.  1  pk.  subtract  4  bu.  5  qt. 

5.  Multiply  30  A.  11  sq.  rd.  by  10. 

6.  Divide  159  A.  29.5  sq.  rd.  by  2. 

7.  How  many  bags  containing  2  bu.  1  pk.  each  can  be 
filled  from  a  bin  of  wheat  containing  256  bu.  2  pk.  ? 

8.  How  many  revolutions  will  a  bicycle  wheel  7  ft. 
4  in.  in  circumference  make  in  traveling  25  mi.  ? 

9.  How  many  times  can  a  bushel  measure  be  filled 
from  a  bin  8  ft.  square  and  6  ft.  deep  ?  Will  there  be  a 
remainder  ? 

10.  How  many  gallons  of  water  will  a  tank  4  ft.  7  in. 
by  2  ft.  11  in.  by  1  ft.  3  in.  contain  ? 

11.  How  many  times  is  7  ft.  6  in.  contained  in  195  mi. 
280  rd.  ? 


METRIC  SYSTEM  OF   WEIGHTS  AND  MEASURES      71 

12.  How  much  coal  is  there  in  three  carloads  of  38  T. 
3  cwt.  41  lb.,  29  T.  7  cwt.  5  lb.,  32  T.  17  cwt.  70  lb.  ? 

13.  Wliat  must  be  the  length  of  a  shed  7  ft.  liigh  and 
9  ft.  wide  to  contain  50  cd.  of  16  in.  wood  ? 

14.  If  a  ton  of  coal  occupies  36  cu.  ft.,  what  must  Ije 
tlie  depth  of  a  bin  6  ft.  wide  by  7 J  ft.  long  in  order  that 
it  may  contain  10  T.  ? 

15.  Divide  320  rd.  4  yd.  by  10  rd.  2  yd. 

16.  How  many  feet  are  there  in  |  of  a  mile  ? 

17.  Reduce  17  pt.  to  a  decimal  of  a  gallon. 

18.  How  many  steps  does  a  man  take  in  walking  a  mile 
if  he  advances  2  ft.  10  in.  each  step  ? 

19.  The  pound  avoirdupois  contains  7000  gr.  Find 
the  greatest  weight  that  will  measure  both  a  pound  Troy 
and  a  pound  avoirdupois.  Find  the  least  weiglit  that 
can  be  expressed  without  fractions  in  both  pounds  Troy 
and  pounds  avoirdupois. 

20.  A  cubic  foot  of  water  weighs  1000  oz.  avoirdupois. 
Find  the  number  of  grains  Troy  in  a  cubic  inch  of  water. 

METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES 

128.  Late  in  the  eighteenth  century  France  invented  the  metric 
system  of  weights  and  measures,  but  it  was  not  made  obligatory  until 
1837.  Previous  to  this  time  there  existed  in  France  the  same  lack  of 
uniformity  in  forming  multiples  and  subniultiples  of  the  units  of 
measure  as  exists  in  our  system  at  the  present  time.  The  metric 
system  is  now  in  use  in  most  civilized  countries  except  the  United 
States  and  England.  It  was  legalized  by  CongTess  in  the  United 
States  in  1866,  but  has  not  been  generally  adopted.  In  scientific 
work  the  system  is  quite  generally  used  in  all  countries, 


MEASUBES 


129.    The   unit    of  length  is  the  meter.      This  is  the 

fundamental  unit,  because   from   it    everv  other  unit   of 

measure  or  weisrht  Is  deriyed ;   hence  the 

/name  metric  system.  The  meter  is  theo- 
'  "^  retically  one  ten-millionth  part  of  the 
/^^  distance  of  the  pole  from  the  equator. 
Though  an  error  has  since  been  dis- 
covered in  the  measurement  of  the  dis- 
tance, the  meter  has  not  been  changed, 
and  a  rod  of  platinum  39.37079  inches 
in  length,  deposited  in  the  archives  at 
Paris,  is  called  the  standard  meter. 


c^ 


CO 


^ — D^ 


-LO 


to 


C\J 


used  to  denote 
means  0.1  of  a 
gram. 


130.  The  unit  of  capacity  is  called 
the  liter.  It  is  a  cube  whose  edge  is 
0.1  of  a  meter. 

131.  The  unit  of  weight  is  the  gram. 
The  gram  is  the  weight  of  a  cube  of 
distilled  water  at  maximum  density, 
whose  edge  is  0.01  of  a  meter. 

132.  The  above  units  of  measure. 
Together  with  the  following  prefixes, 
should  be  carefully  memorized,  because 
from  them  the  whole  metric  system  can 
be  built  up. 

133.  The  Latin  prefixes,  deci.  centi. 
milli,  denote  respectively  0.1,  O.Ml.  O.MOl 
of  the  unit.  The  Greek  prefix  micro  is 
0.000001  of  a  unit.  Thus,  deciuieter 
meter,  and  centicrram   means   0.01   of  a 


METRIC  SYSTEM  OF   WEKIHTS  AND  MEASURES     73 

134.  The  Greek  prefixes,  deea,  hecto,  kilo,  myria,  denote 
respectively  10,  100,  1000,  10000  times  the  unit.  'V\ms, 
Ivilometer  means  1000  meters,  and  hectoliter  means  100 
liters. 

135.  In  general,  nothing  beyond  practice  in  arithmetical 
operations  would  be  gained  in  reducing  from  tlie  metric 
system  to  our  system.  Occasionally,  however,  such  reduc- 
tions are  necessary,  hence,  a  few  of  the  common  equivalents 
are  given  in  the  tables. 

136.    Mkasures  of  Length 

10  uiillinieters  ("'"^)  =  1  centimeter 

10  centimeters  (*^'")  =  1  decimeter 

10  decimeters  (^"'>    =  1  meter  =  80.37  in. 

10  meters  ('")  =  1  decameter 

10  decameters  '^'"^  =  1  hectometer 

10  hectometers  ("'">=  1  kilometer 

10  kilometers  (^"''    =  1  myriameter  (^^""^ 

137.    Square  Measure 

100  square  millimeters  (""-^    =  1  square  centimeter 
100  square  centimeters  ('^^-^  =  1  square  decimeter 
100  square  decimeters  ^^'"-^    =  1  square  meter 
100  square  meters  ('"-^  =  1  square  decameter 

100  square  hectometers  *"'"-^=  1  square  kilometer  ^•^'"-^ 

This  table  may  be  extended  by  squaring  each  unit  of  length  for 
the  corresponding  unit  of  square  measure.  The  denominations  given 
in  tlie  table  are  the  only  ones  in  common  use. 

In  measuring  land,  the  square  decameter  is  called  the  are.  the 
square  hectometer,  the  hectare  =  2.47  acres,  and  the  square  meter,  the 
pentare. 


74  MEASURES 

138.    Cubic  Measure 
1000  cubic  millimeters  ('"'"^>  =  1  cubic  centimeter 
1000  cubic  centimeters  ('^"•^^    =  1  cubic  decimeter 
1000  cubic  decimeters  ^<^'"^)   =  1  cubic  meter  ('"^) 

This  table  may  be  extended  by  cubing  each  unit  of  length  for  the 
corresponding  unit  of  cubic  measure.  The  denominations  given  in 
the  table  are  the  only  ones  in  common  use. 

The  cubic  meter  is  used  in  measuring  wood,  and  is  called  the  stere. 

139.    Measures  of  Weight 

10  milligrams  ('"^^  =  1  centigram 

10  centigrams  ("^s)  =  1  decigram 

10  decigrams  ^^s)  —  1  gram 

10  grams  (e)  =  1  decagram 

10  decagrams  (^^^  =  1  hectogram 

10  hectograms  (Hg)  ==  1  kilogram  =  2.2  lb. 

10  kilograms  (*^s)  =  l  myriagram 

10  myriagrams  '■^s)  =  1  quintal 

10  quintals  (Q)  =  1  tonneau  (T) 

The  metric  ton  or  tonneau  is  the  weight  of  one  cubic  meter  of 
distilled  water  =  2201.62  pounds. 

140.    Measures  of  Capacity 

10  milliliters  C"!)  =  1  centiliter 

10  centiliters  ('^i)  =  1  deciliter 

10  deciliters  <«^)  =  1  liter  =  1  qt.  nearly 

10  liters  (^)  =  1  decaliter 

10  decaliters<Di)  =  1  hectoliter  =  2.837  bu. 

10  hectoliters  ("i)  =  1  kiloliter  (^D 


METRIC  SYSTEM  OF   WEIGHTS  AND  MEASURES     75 

EXERCISE  21 

1.  What  is  the  weight  of  a  liter  of  water  ?  Give  the 
result  ill  grams. 

2.  What  is  the  weight  of  a  cubic  centimeter  of  water? 
of  a  cubic  meter? 

3.  What  is  the  weight  of  15"'  of  water  ? 

4.  Find  the  sum  of  21.14"",  321'  and  1.25'^^  Give 
the  result  in  liters. 

5.  P^ind  in  hectares  and  ares  the  area  of  a  field  450'" 
long  and  200'"  wide. 

6.  If  gold  is  19.36  times  as  heavy  as  water,  find  in  kilo- 
grams the  weight  of  a  bar  of  gold  10"^'^^  long,  30'"'"  wide 
and  25"^'"  thick. 

7.  How  many  square  millimeters  are  there  in  a  square 
centimeter  ?   in  two  meters  square  ? 

8.  Reduce  240064'"'"  to  kilometers,  etc. 

9.  Reduce  3463''^  to  hectares,  etc. 

10.  If  15^^  7^  of  beef  cost  26  francs  37|  centimes,  find 
the  cost  per  kilogram. 

11.  How  many  sacks  will  be  necessary  to  hold  1245"* 
6^'  of  wheat  if  each  sack  holds  1"'  20*  ? 

12.  What  decihial  of  a  decagram  is  6^  4"*^  ? 

13.  How  much  wheat  is  contained  in  1396  sacks,  each 
of  which  contains  1"*  35*? 

14.  If  the  distance  from  the  equator  to  the  pole  is  1000 
myriameters,  how  many  meters  are  there  in  a  degree  ? 

15.  What  will  be  the  price  of  47"^  5^  6o'^  of  land  at 
89.76  francs  per  are  ? 


76  MEASURES 

16.  Mercury  is  13.598  times  as  heavy  as  water.  Find  the 
weight  of  567.859""^  of  mercury. 

17.  If  a  man  steps  80^™  at  each  step,  how  many  steps 
will  he  take  in  walking  10^"'? 

18.  Olive  oil  is  0.914  as  heavy  as  water.  Find  the  cost 
of  a  hectoliter  at  3  francs  a  kilogram. 

19.  A  piece  of  land  1236  meters  square  sold  for  240 
francs  per  hectare.     How  much  did  the  land  bring  ? 

20.  A  person  brought  f  of  a  piece  of  land  containing  2  "* 
15^  at  45  francs  an  are ;  he  sold  |  of  wliat  he  bought  for 
5000  francs.      How  much  did  he  gain  ? 

21.  A  spring  furnishes  5^  of  water  in  2  min.  How 
long  will  it  take  the  spring  to  fill  a  vessel  holding  32| 
liters  ? 

22.  Three  fountains  furnish  3J\  2|^  and  7|^  of  water 
each  minute  respectively.  The  three  together  fill  a  tank 
in  2  hr.  and  43  min.  How  many  hectoliters  of  water  does 
the  tank  contain? 

23.  If  8^^  of  land  are  bought  for  19200  francs  and  sold 
for  25.20  francs  per  square  meter,  how  much  is  gained  by 
the  transaction  ? 

24.  If  sea  water  is  1.026  times  as  heavy  as  distilled 
water  and  olive  oil  is  0.914  as  heavy,  how  much  more  than 
an  equal  volume  of  olive  oil  will  a  hectoliter  of  sea  water 
weigh  ? 

141.  Measures  of  Angles  and  Time.  The  sexagesimal  division  of 
numbers  is  undoubtedly  of  Babylonian  origin.  The  Babylonian 
priests  in  their  astronomical  work  reckoned  the  year  as  360  days. 
They  supposed  the  sun  to  revolve  around  the  earth  once  each  year 
and  hence  divided  the  circumference  of  the  circle  into  860  parts,  each 
of  which  re^jresented   the  apparent  daily  path  of  the   sun.     They 


MEASURES   OF  ANGLES   AND    TIME  11 

probably  knew  the  construction  of  the  regular  hexagon  by  applying 
the  radius  to  the  circumference  six  times.  It  was  then  natural  to 
take  one  of  the  00  parts  thus  cut  off  as  a  unit  and  to  further  subdivide 
this  unit  into  00  equal  parts,  and  so  on,  according  to  their  method  of 
sexagesimal  fractions.  This  is  the  oi'igin  of  our  degree,  minute  and 
second.  The  names  minutes  and  seconds  are  taken  from  the  Latin 
partes  minutcc  prhnee  ^nd  partes  minutcE  secundoi. 

142.  The  principal  measures  of  time  are  the  day  and  the  year.  The 
day  is  the  average  time  in  which  the  earth  revolves  on  its  axis.  The 
division  of  the  day  into  24  hours,  of  the  hour  into  00  minutes,  and 
of  the  minute  into  00  seconds  is  probably  due  to  the  Babylonians. 
The  solar  year  is  the  time  in  w^iich  the  earth  travels  once  around  the 
sun.     It  contains  305.2420  days. 

143.  In  B.C.  40  Julius  C?esar  reformed  the  calendar  and  decreed 
that  there  should  be  three  successive  years  of  305  days  followed  by  a 
year  of  300  days  to  account  for  the  difference  of  0.2420  of  a  day 
between  the  year  of  305  days  and  the  solar  year.  The  difference  be- 
tween four  years  of  305  days  and  four  years  of  305.2420  days  is  only 
0.9704  of  a  day,  so  that  if  a  w^iole  day  is  added  every  fourth  year  there  is 
added  0.0290  of  a  day  too  much.  In  1582  Pope  Gregory  XIII  corrected 
this  by  striking  10  days  from  the  year,  —  calling  Oct.  5th  Oct.  15th, — 
and  he  decreed  that  three  leap  years  were  to  be  omitted  in  every  four 
hundred  years.  Every  year  whose  number  is  divisible  by  four  is  a  leap 
year  unless  it  is  a  year  ending  a  century,  as  1900,  when  it  is  a  leap 
year  only  if  divisible  by  400.  1800,  1900,  2100.  are  not  leap  years,  but 
1000,  2000,  2400,  are.  The  Gregorian  calendar  was  adopted  at  once 
in  Roman  Catholic  countries  and  in  England  in  1752.  At  the  same 
time  in  England  the  beginning  of  the  year  was  changed  from 
]\Iarch  25  to  Jan.  1.  Russia  and  some  other  countries  still  use  the 
Julian  calendar.  Since  1582  they  have  had  three  more  leap  years 
(1700,  1800  and  1900)  than  countries  using  the  Gregorian  calendar, 
and  hence  are  now  13  days  behind  other  countries.  What  we  call 
Jan.  23  is  Jan.  10  with  them. 

144.  Dates  given  according  to  the  Julian  calendar  are  called  Old 
Style  (O.  S.)  and  dates  according  to  the  Gregorian  calendar  are  called 
New  Style  (X.  S.) 


LONGITUDE   AND   TIME 

145.  Longitude  is  distance  east  or  west  of  the  prime 
meridian.  The  meridian  of  the  Royal  Observatory  at 
Greenwich,  England,  is  the  prime  meridian  generally 
adopted,  and  longitude  is  reckoned  east  or  west  180°  from 
that  meridian. 

Since  the  earth  makes  one  complete  revolution  on  its 
axis  in  24  hours,  each  place  in  its  surface  passes  through 
360°  in  that  time.     Hence  : 

360°  of  longitude  corresponds  to  24  hr.  of  time. 
1°  of  longitude  corresponds  to  gl-^  of  24  hr.,  or  ^-^  hr.,  or  4  min. 
1'  of  longitude  corresponds  to  ^  of  4  min.,  or  4  sec. 
1"  of  longitude  corresponds  to  ^^,  of  4  sec,  or  -^^  sec. 
And 

24  hr.  of  time  corresponds  to  .360°  of  longitude. 
1  hr.  of  time  corresponds  to  2V  of  360°,  or  15°. 
1  min.  of  time  corresponds  to  -^^  of  15°,  or  15'. 
1  sec.  of  time  corresponds  to  ^  of  15',  or  15". 

Ex.  1.  The  difference  in  time  between  two  places  is 
2  hr.  25  min.  13  sec.     What  is  the  difference  in  longitude? 

Solution.  2x1.5°  =  30°, 

25  X  15'  =  375'  =  6°  L5', 

13  X  15"  =  19.5"  =  3' 1.5", 

30°  +  6°  15'  +  3'  15"  =  36°  18'  15". 

Check  by  reducing  36°  18'  1.5"  to  hours,  minutes,  seconds,  as  in  the 
following  example. 

78 


LONGITUDE  AND   TIME  79 

Ux.  2.  The  difference  in  longitude  of  two  places  is 
46°  32'  45".     What  is  the  difference  in  time  ? 

Solution.     4G  x  j\  hr.  =  13  hr.  4  min. 

32  X  4  sec.  =  128  sec.  =  2  niin.  8  sec. 

45  X  15  sec.  =  3  sec. 

3  hr.  4  mill.  +  2  iiiin.  8  sec.  +  3  sec.  =  3  hr.  6  niin.  11  sec. 

Check  by  reducing  3  hr.  9  min.  11  sec.  to  degrees, 
minutes  and  seconds  as  in  Ux,   1. 

EXERCISE   22 

1.  In  what  direction  does  the  sun  appear  to  move  as  the 
earth  revolves  on  its  axis  ? 

2.  How  many  degrees  pass  under  the  sun's  rays  in  5  hr.? 

3.  When  it  is  noon  at  Chicago,  what  time  is  it  at  a  place 
15°  15'  east  of  Chicago  ?  45°  30'  45"  west  ? 

4.  What  is  the  difference  in  longitude  between  two 
phxces,  the  difference  in  time  being  1  hr.  4  min.? 

5.  A  person  travels  from  Detroit  until  his  watch  is 
45  min.  fast.  In  what  direction  and  through  how  many 
degrees  has  he  traveled  ? 

6.  What  is  the  difference  in  time  between  two  places 
whose  longitudes  are  75°  and  60°  ? 

7.  When  it  is  9  a.m.  local  time  at  Washington,  it  is 
8  hr.  7  min.  4  sec.  at  St.  Louis ;  the  longitude  of  Washing- 
ton being  77°  2'  W.,  what  is  the  longitude  of  St.  Louis  ? 

146.  International  Date  Line.  Suppose  that  two  men,  starting  from 
the  prime  meridian  on  ^Monday  noon,  travel  the  one  eastward  and  the 
other  westward,  each  traveling  just  as  fast  as  the  earth  rotates.     The 


80 


LONGITUDE  AND   TIME 


man  v;\\o  goes  west  as  fast  as  the  earth  turns  east  keeps  exactly 
beneath  the  sun  all  the  time  ;  and  it  seems  to  him  to  be  still  Monday 
noon  when  he  reaches  his  starting  point  again  twenty-four  hours 
later.  He  has  lost  a  day  in  his  reckoning  by  traveling  westward 
around  the  earth. 

The  other  man  travels  eastward  over  the  earth  as  fast  as  the  earth 
itself  turns  eastward,  and  therefore  he  moves  away  from  the  sun 
twice  as  fast  as  the  prime  meridian  does.  After  twelve  hours'  travel 
he  reaches  the  meridian  of  180°,  but  twelve  hours  rotation  has  carried 
this  meridian  beneath  the  sun,  and  so  the  traveler  reaches  it  at  noon. 
In  twenty-four  hours  the  man  reaches  his  starting  point  on  the  prime 


0IQiQI0iei(DI0IGIQI0 


7  PM         9  PM= 
1 


150         IZOi-°"e  90  West 60 


meridian,  but  twenty-four  hours'  rotation  has  brought  this  meridian 
beneath  the  sun  again,  so  the  traveler  reaches  it  on  the  second  noon 
after  his  start ;  he  therefore  supposes  it  to  be  Wednesday  noon,  though 
really  it  is  but  twenty-four  hours  after  Monday  noon.  He  has  gained 
a  day  in  his  reckoning  by  traveling  eastward  around  the  earth.  To 
correct  such  errors  in  their  dates,  navigators  usually  add  a  day  to  their 
reckoning  when  they  sail  westward  across  the  meridian  of  180°,  and 
subtract  a  day  when  they  cross  it  to  the  eastward.  The  line  where  the 
adjustment  is  made,  corresponding  in  general  with  the  meridian  of 
180°,  is  called  the  international  date  line. 

The  map  represents  the  earth  when  it  is  noon  Feb.  1  at  Greenwich, 


LONGITUDE  AND    TIME  81 

It  is,  therefore,  one  hour  earlier  in  tlie  day  for  each  15°  west  of 
GreeuAvich  and  one  hour  later  in  the  day  for  each  15^  east  of  (ireen- 
Nvich.  Hence  180°  west  of  (Jreenwich  it  is  midnight  of  Jan.  -il,  and 
180°  east  of  Greenwich  it  is  midnight  of  Feb.  1. 

When  it  is  6  a.m.  Feb.  1  at  Greenwich,  at  OO"*  W.  it  is  midnight  of 
Jan.  31,  and  at  90°  E.  it  is  noon  of  Feb.  1;  at  180°  W.  it  is  0  p.m. 
Jan.  31  and  at  180°  E.  it  is  6  p.m.  Feb.  1.  In  this  case  it  is  Jan.  31 
in  all  longitudes  from  90°  W.  westward  to  the  date  line,  and  Feb.  1 
in  all  longitudes  from  90°  W.  eastward  to  the  date  line. 

When  it  is  midnight  of  Jan.  31  at  Greenwich,  at  90°  W.  it  is  6  p.m. 
Jan.  31  and  at  90  E.  it  is  6  a.m.  Feb.  1 ;  at  180°  W.  it  is  noon  Jan.  31 
and  at  180°  E.  it  is  noon  Feb.  1.  In  this  case  it  is  Jan.  31  in  all 
longitudes  from  Greenwich  westward  to  the  date  line,  and  Feb.  1  in 
aU  longitudes  from  Greenwich  eastward  to  the  date  line. 

When  it  is  6  p.m.  Jan  31  at  Greenwich,  at  90°  AV.  it  is  noon  Jan.  31, 
and  at  90°  E.  it  is  midnight  Jan.  31;  at  180°  W.  it  is  6  a.m.  Jan.  31, 
and  at  180°  E.  it  is  6  a.m.  Feb.  1.  In  this  case  it  is  Jan.  31  in  all 
longitudes  from  90°  E.  westward  to  the  date  line,  and  Feb.  1  in  all 
longitudes  from  90°  E.  eastward  to  the  date  line. 


1.  When  it  is  noon  February  1  at  Greenwich,  what 
date  is  it  at  Paris  ?  at  New  York  City  ?  at  San 
Francisco  ? 

2.  Imagine  the  midnight  line  of  Jan.  31  as  a  dark  line 
moving  westward  parallel  to  the  meridian.  Everywhere 
on.  this  line  it  is  midnight.  Behind  this  line  it  is  Feb.  1 ; 
in  front,  Jan.  31.  On  what  part  of  the  earth's  surface 
is  it  Feb.  1,  and  on  what  part  Jan.  31  when  this  im- 
aginary midnight  line  has  reached  the  prime  meridian? 
90°  E.?     140°  E.? 

3.  What  date  will  be  in  front  of  this  line  when  it 
reaches  180°  ?     What  date  will  be  behind  it  ? 

lt3ian's  adv.  ar. — 6 


82 


LOSGITUDE  AXD    TIME 


4.  After  crossing  the  180th  meridian  and  passing  on 
to  175°  E..  what  date  is  before  the  line  and  what  date 
behind  it  ? 

5.  What  change  must  be  made  in  the  calendar  of  a 
ship  crossing  this  line  going  westward  ?  Going  east- 
ward ? 


147.    Table  of  longitudes  for  use  in  solving  problems : 


Ann  Arbor.  Mich. 

Albany,  N.Y.  .  . 

Boston     .     .  .  . 

Berlin      .     .  .  . 

Brussels  .     .  .  . 

Chicago  ,     .  .  . 

Cincinnati  .  .  . 
Cambridge,   Eng. 

Cape  Town .  .  . 

Calcutta .     .  .  . 

Detroit    .     .  .  . 

Dublin     .     .  .  . 

Honolulu     .  .  . 


S3°43'4S"  W. 
730  44'  4S"  W. 

7V    3'  30"  W. 
13°  23'  43"  E. 

4°  22'  9'E. 
87°  36' 42"  W. 
84^26'    0"W. 

0°  5'41"E. 
18°  28'  45"  E. 
88°  19'  2"E. 
83°    5'    7"W. 

6°   2'30"W. 
157°  52'    0"W. 


London .     . 
Lisbon  .     . 
^Melbourne 
Xew  Orleans 
Xew  York  , 
Paris      .     . 
Peking  .     . 
Rome     .     . 
.San  Francisco 
St.  Louis    . 
Sydney  .     . 
Tokyo    .     . 
Washino-tun 


0=    5'38"E. 
9^  11'  10"  W. 
144°  58'  42"  E. 
90°   3'28"W. 
74°    0'    3"^y. 
2'  2U'  15"  E. 
116°  26'    0"E. 
12°  27'  14"  E. 
122°  26' 15"  W. 
90M2  11' W. 
151°  11'    0"E. 
139°  42' 30  "E. 
77°    1'        W. 


EXERCISE   23 

1.  Determine  the  time  and  date  at  Ann  Arbor,  Berlin, 
Cape  Town  and  Peking  when  it  is  midnight  May  15  at 
Greenwich. 

2.  When  it  is  noon  March  1  at  Rome,  what  time  and 
date  is  it  at  San  Francisco?   at  Sydney?  at  Detroit? 


LONGITUDE  AND   TIME 


83 


3.  If  a  man  were  to  travel  westward  around  the  earth 
in  121  da.,  in  how  many  days  would  he  actually  make  the 
trip  by  the  local  time  of  the  places  he  passes  tlirough? 
In  how  many  days  would  lie  make  the  trip  traveling  east- 
ward ? 

4.  When  it  is  noon  Sunday,  Jan.  31,  on  the  90th  me- 
ridian west,  what  part  of  the  world  has  Sunday  ?  What 
is  the  day  and  date  on  the  other  part  ? 

5.  When  it  is  3  p.m.  Feb.  5  on  the  45th  meridian  east, 
what  part  of  the  world  has  Feb.  5,  and  what  is  the  date 
on  the  other  part  ? 


^  #      /        ^'/r-~.^  prandon;     .    ..:  ^     Fort 


STANDARD  TEVIE 
BELTS 


148.  Standard  Time.  In  order  to  secure  uniform  time 
over  considerable  territory,  in  1883  the  railroad  companies 
of  the  United  States  and  Canada  decided  to  adopt  standard 
time.  They  divided  the  country  into  four  time  belts,  each 
of  approximately  15°  of  longitude  in  width.  The  time  in 
the  various  belts  will  therefore  differ  by  hours,  while  the 


84  LONGITUDE  AND   TIME 

minute  and  second  hand  of  all  timepieces  will  remain 
the  same.  The  correct  time  is  distributed  by  telegraph 
throughout  the  United  States  from  the  Naval  Observa- 
tory at  Washington  each  day. 

149.  The  Eastern  time  belt  lies  approximately  7J°  each 
side  of  the  75th  meridian,  and  has  throughout  the  local 
time  of  the  75th  meridian.  Similarly,  the  Central,  Moun- 
tain and  Pacific  time  belts  lie  approximately  7|°  each  side 
of  the  90th,  105th  and  120th  meridians,  and  the  time 
throughout  in  each  belt  is  determined  by  the  local  time 
of  the  90th,  105th  and  120th  meridians. 

150.  These  divisions  are  not  by  any  means  equal  or 
uniform,  since  railroads  change  their  time  at  important 
junctions  and  termini.  Consequently,  variations  are  made 
from  the  straight  line  to  include  such  places.  Thus,  while 
most  roads  change  from  Eastern  to  Central  time  at  Buffalo, 
the  Canadian  roads  extend  the  Eastern  belt  much  farther 
west. 

151.  Standard  time  has  more  recently  been  adopted  by 
most  of  the  leading  governments  of  the  world.  With 
few  exceptions  the  standard  meridian  chosen  represents  a 
whole  number  of  hours  from  the  prime  meridian  through 
Greenwich. 

Great  Britain,  Belgium  and  Holland  use  the  time  of  the  meridian 
through  Greenwich.  France  uses  the  time  of  the  meridian  (2°  20'  E.) 
through  Paris.  Cape  Colony  uses  the  time  of  the  meridian  22°  30'  E. 
Germany,  Italy,  Austria,  Denmark,  Norway  and  Sweden  use  the  time 
of  the  15th  meridian  east.  Roumania,  Bulgaria  and  Natal  use  the 
time  of  the  30th  meridian  east.  Western  Australia  uses  the  time  of 
the  120th  meridian  east.  Southern  Australia  and  Japan  use  the  time 
of  the  135th  meridian  east.  Eastern  Australia,  Victoria  and  Queens- 
land and  Tasmania  use  the  time  of  the  loOtli  meridian  east.  New 
Zealand  uses  the  time  of  170"^  30'  E. 


LONGITUDE  AND   TIME  85 

EXERCISES   24 

1.  Wlien  it  is  7  P.isr.  at  IMiihidelpliia,  what  time  is  it  at 
London  ?    at  Paris  ?    at  Berlin  ? 

2.  What  is  the  difference  in  time  between  Boston  and 
Rome  ?     Melbourne  and  Tokyo  ? 

3.  It  is  7  A.M.  INIarch  1  at  St.  Louis.     What  is  the  time 
and  date  at  Tokyo  ? 

4.  When  it  is  1.30  i*.]\r.  at  Buffalo,  what  time  is  it  at 
Cleveland  ? 

5.  What  is  the  difference  between  the  local  and  standard 
time  of  Boston  ?  of  Chicago  ?  of  St.  Louis  ?  of  New  York? 

6.  The  local  time  of  Detroit  is  27.658  min.  faster  than 
the  standard  time.     Find  the  longitude  of  Detroit. 

7.  When  it  is  noon  local  time  at  Boston,  what  is  the 
standard  time  at  Ogden,  Utah? 


THE   EQUATION 

152.  An  equation  is  a  statement  of  equality  of  two  num- 
bers or  expressions.  Thus,  5  =  3+2  and  3  x  5  =  15  are 
equations. 

153.  The  equation  is  one  of  the  most  powerful  in- 
struments used  in  mathematics  and  can  be  used  to 
decided  advantage  in  the  solution  of  many  arithmetical 
problems. 

154.  The  book  written  by  the  Egyptian  priest  Ahmes,  and  re- 
ferred to  elsewhere,  is  one  of  the  very  oldest  records  of  the  extent  of 
mathematical  knowledge  among  the  ancients.  In  this  book  we  find 
a  very  close  relation  between  arithmetic  and  algebra.  A  number  of 
problems  are  given  leading  to  the  simple  equation.  Here  the  unknown 
quantity  is  called  hau  or  "  heap,"  and  the  equation  is  given  in  the  fol- 
lowing form:  heap  its  |,  its  J,  its  \,  its  whole,  gives  33,  or  '{x  -\-  Ix 
■\-\  X  -\-  X  =  33.  (Cajori,  "  History  of  Elementary  Mathematics," 
p.  23.) 

Ex.  1.  Find  a  number  such  that,  if  30  is  subtracted, 
I  of  the  original  number  will  remain. 

The  given  relation  may  be  expressed  as  follows : 

The  number  diminished  by  30  equals  |  of  the  number. 

From  this  relation  we  are  to  find  the  number. 

Let  X  =  the  number. 

Then  a:  —  30  =  the  number  diminished  by  30,  and  |  x  =  |  of  the 
number. 

.-.  X  —  30  =  I  a:  is  the  equation  expressing  the  relation  between  the 
number  given  in  the  conditions  of  the  problem  and  the  letter  repre- 
senting the  number  to  be  found. 


THE  EQUATION 


87 


Solution.  Since  we  wish  to  obtain  an  equation  with  x  alone  on  the 
left  side  and  only  numerical  quantities  on  the  right  side,  we  proceed 
as  follows : 

Subtracting  |  x  from  both  sides  of  the  equation,  we  have 

X  -  30  -  I  X  =  I  a;  -  I  z,  Axiom  3. 

or  \  a;  -  30  =  0. 

Adding  30  to  both  sides  of  the  equation,  we  have 

i  a:  -  30  +  30  =  30,  Axiom  2. 

or  \x  =  30. 

Multiplying  both  terms  by  3,  we  have 

a;  =  90. 

.-.  90  is  the  required  number. 

To  check  the  result,  substitute  90  for  x  in  the  original  equation. 

This  gives  us  ,  ^^ 

^  90  -  30  =  f  of  90, 

or  60  =  60. 

.-.  X  =  90  is  the  correct  result. 


Ex.  2.    The  sum  of  two  numbers  is  20  and  their  differ- 
ence is  4.     What  are  the  numbers  ? 

Solution.     Let  x  =  the  larger  number. 

.  Then  a:  —  4  =  the  smaller  number. 

.'.  X  -h  X  -■^  =  20. 

2  X  =  24,  Axiom  2. 

or  X  =  1'2  =  the  larger  number.  Axiom  5. 

Check.     12  —  4  =  8  =  the  smaller  number. 
12  +  8  =  20.     12  -  8  =  4. 


THE  EQUATION 


EXERCISE   25 


1.  The  sum  of  two  numbers  is  31  and  their  difference 
is  11.     What  are  the  numbers  ? 

2.  The  difference  of  two  numbers  is  60,  and  if  both 
numbers  are  increased  by  5  the  greater  becomes  four  times 
as  large  as  the  smaller.     What  are  the  numbers  ? 

3.  Find  two  numbers  such  that  their  difference  is  95 
and  the  smaller  divided  by  the  greater  is  |. 

4.  After  spending  ^  of  his  money  a  man  pays  bills  of 
125,  140  and  |14,  and  finds  that  he  has  $132  left.  How 
much  money  had  he  at  first  ? 

5.  Find  a  number  such  that  its  half,  third  and  fourth 
parts  shall  exceed  its  fifth  part  by  106. 

6.  A  father  wishes  to  divide  f  28,000  among  his  two 
sons  and  a  daughter  so  that  the  elder  son  shall  receive 
twice  as  much  as  the  younger,  and  the  younger  son  tAvice 
as  much  as  the  daughter.     Find  the  share  of  each. 


POAYERS   AND   ROOTS 

155.  Archimedes  ('287-212  n.c.)  in  his  measurenients  of  the  circle 
conii)ated  the  approximate  value  of  a  number  of  square  roots,  but 
nothing  is  known  of  his  method.  A  little  later,  Heron  of  Alexandria 
also  used  the  approximation  Va^  ^  ^  =  a  -\-  —,  e.g.  V85  =  V9'-^  +  4 
=  9  +  A.  ^  "^ 

156.  The  Hindus  included  powers  and  roots  among  the  funda- 
mental processes  of  aritlimetic.  As  early  as  476  a.d.  they  used  the 
formulai  (a  +  by  =  a'^  +  2  ab  +  b'^  and  («  +  by  =  (fi  +  3  uH)  4-  3  ab"^  +  b^ 
in  the  extraction  of  square  and  cube  root,  and  they  separated  numbers 
into  periods  of  two  and  three  figures  each. 

157.  The  Arabs  also  extracted  roots  by  the  formula  for  (a  +  by. 
They  introduced  a  radical  sign  by  placing  the  initial  letter  of  the 
word  Jird  (root)  over  the  number. 

158.  The  power  of  a  number  is  the  product  that  arises 
by  multiplying  the  number  by  itself  any  number  of  times. 
The  second  power  is  called  the  square  of  the  number  and 
the  number  itself  is  called  the  square  root  of  its  second 
power.  The  third  power  is  called  the  cube  of  the  number, 
and  the  number  itself  is  called  the  cube  root  of  its  third 
power.  Thus,  4  is  the  square  of  2,  and  2  is  the  square  root 
of  4.     125  is  the  cube  of  5,  and  5  is  the  cube  root  of  125. 

159.  Since  the  square  root  of  a  number  is  one  of  the  two 
equal  factors  of  a  perfect  second  power,  numbers  that  are 
not  exact  squares  have  no  square  roots.  However,  they 
are  treated  as  having  approximate  square  roots.  These 
approximate  square  roots  can  be  found  to  any  required 

89 


90  POWERS  AND  ROOTS 

degree  of  accuracy.     Thus,  the  square  root  of  91  correct 
to  0.1  is  9.5,  and  correct  to  0.01  is  9.54. 

160.  According  to  the  definition,  only  abstract  numbers 
have  square  roots. 

161.  The  square,  the  cube,  fourth  power,  etc.,  of  2  are 
expressed  by  2^,  2^,  2^,  etc.  The  small  figure  which  de- 
notes the  power  is  called  the  exponent. 

162.  The  square  root  is  denoted  by  the  symbol  Vi  the 
cube  root  by  -^,  the  fourth  power  by  ^y,  etc. 

163.  The  following  results  are  important : 

12=1, 

102=100, 
1002=10000, 
10002=1000000. 

The  squares  of  all  numbers  between  1  and  10  lie  be- 
tween 1  and  100,  the  squares  of  all  numbers  between  10 
and  100  lie  between  100  and  10000,  etc.  Hence,  the  square 
of  a  number  of  one  digit  is  a  number  of  one  or  two  digits, 
the  square  of  a  number  of  two  digits  is  a  number  of  three 
or  four  digits,  etc.  It  will  be  noticed  that  the  addition  of 
a  digit  to  a  number  adds  two  digits  to  the  square. 

164.  13=1, 
103=1000, 

1003=1000000,  etc. 

It  will  be  noticed  in  this  case  that  the  addition  of  a  digit 
to  a  number  adds  three  digits  to  its  cube. 


POWERS  AND   HOOTS  91 

165.  0.12  =  0.01, 
0.012=0.0001, 

0.0012  =  0.000001,  etc. 

Hence,  the  square  of  a  decimal  nunibcr  contains  twice  as 
many  digits  as  the  number  itself. 

166.  0.13=0.001, 
0.013=0.000001, 

0.0013=0.000000001,  etc. 

Hence,  the  cube  of  a  decimal  number  contains  three  times 
as  many  digits  as  the  number  itself. 

167.  Laws  of  exponents  : 

Since  '2~  =  2  x  2  and  2^  =  2  x  2  x  2, 

then  2^  X  23  =  2  X  2  X  2  X  2  X  2  =  2^. 

This  may  be  written  in  the  form 

02  X   03  _  02+3  —  05. 

Or,  in  general,  a'"  x  a"  =  a'"+".  I. 

Also, 
(23)4  =  (2  X  2  X  2)  (2  X  2  X  2)  (2  x  2  x  2)  (2  x  2  x  2)  =  23+3+3+3  ^  oi^. 
This  may  be  written  in  the  form 

(23)**  =  23X-*  =  2^2. 

Or,  in  general,  (a'")"  =  a'^'X".  II. 

Also,  25  -  23  =  -x-x-x-x-  ^  o  X  2  =  22. 

2x2x2 

This  may  be  written  in  the  form 

25  _^  03  _  25-3  =  2^. 


92  POWERS  AND  BOOTS 

Or,  in  general,  a"'  -=-  a"  =  a"'-".  III. 

92  92 

Also,  since        —  =  22-2  =  00  (Principle  III),  and  ~  =  1,  .-.  2^  =  1. 

Or,  since  2^  x  2^  =  22+o  =  2'-  (Principle  I),  and  22  x  1  =  22,  .-.  2o  =  l. 

Or,  in  general,  a-'  =  l.  IV. 

Also,  since  ^  =  21-2=2-1  (Principle  III),  and  ^=^  =  ^,  .-.2-^=-. 
22  22     4     2  2 

Or,  since     22  x  2-^  =  2  (Principle  I),  and  22  x  -  =  2,  .-.  2-i  =  i- 

Or,  in  o'eneral,  a~"  =  —  V. 

That  is,  any  number  with  a  negative  exponent  is  equal  to  the  recip- 
rocal of  the  same  number  with  a  numerically  equal  positive  exponent. 

Also,    (22)2  ^  2,  since  (2^)2  =z  2^  x  2^  =  2^  +  *  =  2i  =  2. 

1         .- 
.-.  22  =  v2  (extracting  the  square  root  of  both   members  of  the 

equation  (2^)2  =  2). 

And  (2^)3  =  22,  or  2^  =  V¥K 

Or,  in  general, 

m  m 

{a'^y  =  a'",  or  a'"  =Va^.  VI. 

Ux.    Show  that   1=31=3;    |=30  =  1;    |=3-i  =  i; 

10-1=0.1;   10-2=0.01;   10-5=0.00001. 

,2       .       .i_2       .       ^^^_97^ 


Show  that  8^  =  4;    27^  =  9;   2^=8;  81^  =  27;   32^=6-4. 


EXERCISE   26 

1.  How  many  figures  are  there  in  the  square  of  a  num- 
ber of  3  figures  ?  of  4  figures  ? 

2.  How  many  figures  are  there  in  31^  ?  in  32^  ? 


POWERS  AND  BOOTS  93 

3.  How  many  figures  are  there  in  the  cube  of  a  num- 
ber of  2  figures  ?  of  3  hgures  ? 

4.  How  many  figures  are  there  in  30^?  in  32^? 

5.  How  many  figures  are  there  in  tlie  fourth  power  of 
a  number  of  3  figures  ?  in  the  fifth  power  of  a  number  of  2 
figures  ? 

6.  How  many  figures  are  there  in  the  cube  of  a  num- 
ber of  5  figures  ? 

7.  How  many  figures  are  there  in  the  cube  of  a  num- 
ber of  4  figures  ? 


8.    How    many    figures     are     there     in    ^5929  ?     in 


V1038361  ? 


9.    How  many  figures  are  there  in  VO.04?  in  VO.36 


10.    How    many    figures    are    there    in    V'37.21?     in 

V4.8841? 


11.    How  many  figures  are  there  in  v  1030301  ? 


12.  How  many  figures  are  tliere  in  V 10793861  ? 

13.  Show  by  multiplication  that  (^a-\-hy=a^-{-2ah-{-P, 
and  by  use  of  this  formula  square  32  and  65. 

14.  Show  by  multiplication  that  (a  +  ^  )^  =  a''^  +  3  a^  + 
3  ab^  -f  J3,  and  by  use  of  the  formula  cube  41  and  98. 

15.  Square  234,  first  considering  it  as  230  +  4  and  then 
as  200  +  34. 

16.  Show  that  no  number  ending  in  2,  3,  7  or  8  can  be 
a  perfect  square. 

17.  Prove  that  the  cube  of  a  number  may  end  in  any 
digits. 


94  POWERS  AND  BOOTS 

168.  Square  Root.  The  square  root  can  often  be  de- 
termined by  inspection  if  the  number  can  readily  be  sepa- 
rated into  prime  factors.     Thus  : 

32400  =  24  X  34  X  52. 


.-.  V32400  =  22  X  32  X  5  =  180. 

Ux.  By  separating  into  prime  factors  find  the  square 
root  of  (a)  64,  (5)  17424,  (c)  7056,  (d)  99225,  (e)  680625, 
(/)  2800625,  (c/)  11025,  (A)  81,  (z)  1764,  (j)  9801. 

169.  Since  432  =(40 +  3)2  =  402+  2  x  40  x  3  +  32=  1849, 
by  reversing  the  process  we  can  find  V1849. 

402  +  2  X  40  X  3  +  32  I  40 +  3 
402 


2  X  40  +  3 


2  X  40  X  3  +  32 
2  X  40  X  3  +  32 


170.  In  the  above  we  notice  that  40  is  the  square  root  of  the  first 
part.  After  subtracting  402  the  remainder  is  2  x  40  x  3  +  32.  The 
trial  divisor,  2  x  40,  is  contained  in  the  remainder  3  times.  By  adding 
3  to  2  X  40  the  complete  divisor,  2  x  40  +  3.  is  formed.  The  complete 
divisor  is  contained  exactly  3  times  in  the  remainder.  By  this  division 
the  3,  the  second  figure  of  the  root,  is  found. 

171. 


The  above  is  equivalent  to  the  following  : 

4  3 

1849 

402  = 

1600 

2  X  40  +  3 

249 

=  83 

249 

Separating  1849  into  periods  of  two  figures  each  (why?)  we  find 
that  1600  is  the  greatest  square  in  1800.  4  is  therefore  the  first  figure 
in  the  root.  Subtracting  1600  and  using  2  x  40  as  the  trial  divisor, 
the  next  figure  in  the  root  is  found  to  be  3.  Completing  the  divisor 
by  adding  the  3,  it  is  found  to  be  exactly  contained  in  the  remainder. 
If  the  number  contains  more  than  two  periods,  the  process  is  repeated. 


POWERS  AND   ROOTS  95 

172.  The  whole  process  of  extracting  the  square  root  of  a 
nuiiil^er  is  contained  in  the  formula  (^a-\-b)^  =  a^ -{-2(11^  +  6^. 
The  process  of  extracting  the  cube  root  is  contained  in 
(rt  +  by  =  a^  -h  3  a^b  +  8  ab'^  +  b^.  In  general,  the  process 
of  extracting  any  root  can  be  derived  from  the  correspond- 
ing power  of  {a  +  b). 

Ex.    Extract  the  square  root  of  4529.29. 

The  a  of  the  fonmila  will  always  represent  tlie  part  of  the  root 
ah-eady  found  aud  the  h  the  next  figure. 

6  7    3 
(a  +  hy  =  rt2  +  2  ffj,  +  ]f.  4ry29.29 


a^ 

=  3600 

2  rt  =  120 

929.29 

h=      7 

(2  rt  +  h)b  =  127  X  7  = 

889. 

2rtz=134 

40.29 

b  =  0.3 

(2  fl  +  b)h  =  134.3  X  0.3  = 

40.29 

In  practice  tlie  work  may  be  arranged  as  follows 

6  7  3 
4529.29 
36 


120                  929. 
127                  889. 
134                    40.29 
134.3 40.29 

EXERCISE   27 

1.  In    extracting    the    square    root,    why    should    tlie 
number  be  separated  into  periods  of  two  figures  each  ? 

2.  Where  do  you  begin  to  separate  into  periods  ? 

3.  Separate  each  of  the  following  into  periods :    312, 
4.162,  0.0125,  30000.4. 


96  POWERS  AND   BOOTS 

4.  Will  the  division  by  2  a  always  give  the  next  figure 
of  the  root  ? 

5.  Why  is  2  a  called  the  trial  divisor  ? 

6.  Why  is  2a-\-b  called  the  complete  divisor  ? 

7.  In  the  above   example  explain  how  2  a  can  equal 
both  120  and  134. 

8.  Explain  how  2  ab  +  P  can  equal  both  889  and  40.29. 

9.  Show  how  square  root  may  be  checked  by  casting 
out  the  9's. 

10.  Extract  the  square  root  of  each  of  the  following, 
using  the  formula:  (a)  2916,  (b)  5.3861,  (c)  65.61, 
(d)  0.003721,  (e)  1632.16,  (/)  289444,  (^)  0.597529, 
(A)  103.4289,    (0  978121. 

11.  Extract  the  square  root  of  14400. 

12.  By  first  reducing  to  a  decimal,  extract  the  square 
root  of  |,  H,  |,  If  6|. 

13.  By  first  making  the  denominator  a  perfect  square, 
extract  the  square  root  of  |,  |,  ^'^3,  ^j. 

?  =  10;    hence  J  =  J^-  =  ^:1^  =  0.632^. 
5     25  ^5      ^'25  5 

14.  Which  of  the  two  methods  given  in  Ex.  12  and  13 
is  to  be  preferred  ? 

15.  Extract  to  0.01  the  square  root  of  16,  1.6,  0.016. 

173.   Cube  Root. 

Since  373=  (30  +  7)^  =  30^  +  3  x  30^  x  7  +  3  x  30  x  7^  +  7^  =  50G53, 
by  reversing  the  process  we  can  find  v50653. 

303  +  3  X  302  X  7  +  3  X  30  X  72+78(30  +  7 
303 


3  X  302  +  3  X  30  X  7  +  7' 


3  X  302  X  7  +  3  X  30  X  72  +  78 
3x302x7  +  3x30x72  +  78 


POWERS   AND   JiOOTS  97 

174.  In  the  preceding  we  notice  that  30  is  the  cube  root  of  the  first 
part. 

After  subtracting  30^  the  remainder  is  3  x  SO^x  7  +  3  x  30x  V^  +  T^. 

The  trial  divisor,  3  x  30^,  is  contained  in  the  remainder  7  times. 

By  adding  3  x  30  x  7  +  7'^  the  complete  divisor  is  formed.  This 
complete  divisor  is  contained  7  times  in  the  remainder.  By  this 
division  the  7,  the  second  figure  in  the  root,  is  found. 

175.  The  above  is  equivalent  to  the  following : 

3    7 

50653 

303  =  27000 


23653 

3  X  30^  +  3  X  30  X  7  +  7^  =  3379 

3379  X  7  =  23653 

Separating  50653  into  periods  of  three  figures  each  (Why?)  we 
find  that  27000  is  the  greatest  cube  in  50000.  The  first  figure  of  the 
root  is  therefore  3.  Subtracting  27000  and  using  3  x  30^  as  tlie  trial 
divisor,  the  next  figure  in  the  root  (since  8,  when  the  trial  divisor  is 
completed,  proves  to  be  too  large)  is  found  to  be  7.  Completing  the 
divisor  by  adding  3  x  30  x  7  +  7^,  it  is  found  to  be  contained  exactly 
7  times  in  the  remainder.  If  the  number  contains  more  than  two 
periods,  the  process  is  to  be  repeated. 

Ux.    Extract  the  cube  root  of  362467.097. 

As  in  extracting  the  square  root  of  a  number,  the  a  of  the  formula 
will  always  represent  the  part  of  the  root  already  found  and  the  b  the 
next  figure. 

7     1.     3 
(a  +  hy  =  a^  +  3  a^h  +  3  ah^-  +  h^       362467.097 


a^  = 

313000 

3  a^  =  U700 

19467.097 

b=          1 

(:3  «-2  +  :5 ,,/,  + //2)  =  14911 

14911 

3a-^=15123 

4556.097 

b  =          0.3 

(3  rt2  +  3  rt/,  +  j2)  =  15186.99 

4556.097 

lyman's  adv.  ar.  — 7 


98  POWERS  AND   HOOTS 

In  practice  the  work  may  be  arranged  as  follows : 

7     1.     3 

362467.097 
343 

14700  19467. 

14911 14911. 

15123  4556.097 

15186.99  4556.097 


EXERCISE    28 

1.  Separate  each  of  the  following  numbers  into  periods: 
2500,  2.5,  3046.2971,  0.0125,  486521.3. 

2.  In  extracting  the.  cube  root  of  208527857,  does 
the  division  by  3  a^  give  the  second  figure  of  the  root 
correctly  ?  Why  is  3  a^  called  the  trial  divisor  ?  Why  is 
3  a^  +  3  aJ  +  52  called  the  complete  divisor  ? 

3.  In  the  above  example  explain  how  3  a^  can  equal 
both  14700  and  15123.  Explain  how  S  a^ -^  S  ab -{- b'^  can 
equal  both  14911  and  15186.99. 

4.  Show  how  cube  root  may  be  checked  by  casting 
out  the  9's. 

5.  Extract  the  cube  root  of  each  of  the  following, 
using  the  formula:  (a)  472729139,  (6)  278.455077. 
(c)  1054.977832,   (d)  19683,  (e)  205379,  (/)  25153.757. 

6.  Extract  the  cube  root  of  iff  J,  |^|,  ^f  |||. 

7.  By  reducing  to  a  decimal,  extract  the  cube  root  of 

2     4     5  1     1(^2     QQl 
3^   9'    55'   -'^'-'3'  ^^3* 

8.  By  first  making  the  denominator  a  perfect  cube, 
extract  the  cube  root  of  |-,  ^^,  f,  |. 

9.  Find  correct  to  0.01  the  cube  root  of  12.5,  125,  1.25. 


MENSURATION 

176.  Certain  measurements  have  been  in  very  common  use  in  the 
development  of  arithmetical  knowledge  from  the  earliest  times. 

177.  The  Babylonians  and  Egyptians  used  a  great  variety  of 
geometrical  figures  in  decorating  their  walls  and  in  tile  floors.  The 
sense  perception  of  these  geometrical  figures  led  to  their  actual 
measurement  and  finally  to  abstract  geometrical  reasoning. 

178.  The  Greeks  credited  the  Egyptians  with  the  invention  of 
geometry  and  gave  as  its  origin  the  measurement  of  plots  of  land. 
Herodotus  says  that  the  Egyptian  king,  Sesostris  (about  1400  B.C.), 
divided  Egypt  into  equal  rectangular  plots  of  ground,  and  that  the 
annual  overflow  of  the  Nile  either  washed  away  portions  of  the  plot 
or  obliterated  the  boundaries,  making  new  measurements  necessary. 
These  measurements  gave  rise  to  the  study  of  geometry  (from  ge, 
earth,  and  metron,  to  measure). 

179.  Ahmes,  in  his  arithmetical  work,  calculates  the  contents  of 
barns  and  the  area  of  squares,  rectangles,  isosceles  triangles,  isosceles 
trapezoids  and  circles.  There  is  no  clew  to  his  method  of  calculating 
volumes.  In  finding  the  area  of  the  isosceles  triangle  he  multiplies  a 
side  by  half  of  the  base,  giving  the  area  of  a  triangle  w^hose  sides  are 
10  and  base  4  as  20  instead  of  19.6,  the  result  obtained  by  multiply- 
ing the  altitude  by  half  of  the  base.  For  the  area  of  the  isosceles 
trapezoid  he  multiplies  a  side  by  half  the  sum  of  the  parallel  bases, 
instead  of  finding  the  altitude  and  multiplying  that  by  half  the  sum 
of  the  two  parallel  sides.  He  finds  the  area  of  the  circle  by  subtract- 
ing from  the  diameter  i  of  its  length  and  squaring  the  remainder. 
This  leads  to  the  fairly  correct  value  of  3.1604  for  tt. 

180.  The  Egyptians  are  also  credited  with  knoM'ing  that  in  special 
cases  the  square  on  the  hypotenuse  of  a  right  triangle  is  equal  to  the 
sum  of  the  squares  of  the  other  two  sides.  They  were  careful  to 
locate  their  temples  and  other  public  buildings  on  north  and  south, 

99 


100 


MENSURATION 


and  east  and  west  lines.  The  north  and  south  line  they  determined 
by  means  of  the  stars.  The  east  and  west  line  was  determined  at 
right  angles  to  the  other,  probably  by  stretching  around  three  pegs, 
driven  into  the  ground,  a  rope  measured  into  three  parts  which  bore 
the  same  relation  to  each  other  as  the  numbers  3,  4  and  5.  Since 
32  _^  42  _  52^  ^ijjg  gave  the  three  sides  of  a  right  triangle. 

181.  The  ancient  Babylonians  knew  something  of  rudimentary 
geometrical  measurements,  especially  of  the  circle.  They  also  ob- 
tained a  fairly  correct  value  of  tt. 

182.  It  was  the  Greeks  who  made  geometry  a  science  and  gave 
rigid  demonstrations  of  geometrical  theorems. 

183.  The  importance  of  certain  measurements  gives 
mensuration  a  prominent  place  in  arithmetic  to-day.  The 
rules  and  formulae  of  the  present  chapter  will  be  developed 
without  the  aid  of  formal  demonstration. 


184.  The  Rectangle.  If  the 
unit  of  measure  CYXN  is  1  sq. 
in.,  then  the  strip  CYMD  con- 
tains 5x1  sq.  in.  =  5  sq.  in., 
and  the  whole  area  contained  in 
the  three  strips  will  be  3  x  5 
sq.  in.,  or  15  sq.  in. 


M 


D 


X 


B 


N 


185.  The  dimensions  of  a  rectangle  are  its  base  and  alti- 
tude, and  the  area  is  equal  to  the  product  of  the  base  and 
altitude.  That  is^  the  7iumber  of  square  units  in  the  area  is 
equal  to  the  product  of  the  numbers  that  represeiit  the  base 
and  altitude.  If  we  denote  the  area  by  A^  the  number  of 
units  in  the  base  by  b,  and  the  numbers  of  units  in  the  alti- 
tude by  a,  then  A  =  ab  and  any  one  of  the  three  quantities, 
A^  5,  «,  can  be  determined  when  the  other  two  are  given. 

Thus,  if  the  area  of  a  rectangle  is  54  sq.  in.  and  the  altitude  is  6  in., 
the  base  can  be  determined  from  Qh  =  54,  or  h  —  9  in. 


MENSURATION 


101 


/ 

/m 

h 

'/ 

a 

o 

A 

' 

R' 

D' 

N 

/ 

/„ 

h 

C 

1 

186.  If  the  dimensions  of  a  rectangle  are  equal,  the 
figure  is  a  square  and  the  area  is  equal  to  the  second 
power  of  a  number  denoting  the  length  of  its  side,  or 
A  =  a^.  For  this  reason  the  second  power  of  a  number  is 
called  its  square. 

187.  The  Parallelogram.  Any  parallelogram  has  the 
same  area  as  a  rectangle  with  tlie  same  base  and  altitude, 
as  can  be  shown  by  dividing  the  parallelogram  ABCD 
into  two  equal  parts,  M  and  iV,  ^  ^ 
and  placing  them  as  in  A'B'  O'D'^ 
thus  forming  a  rectangle  with  the 
same  base  and  altitude  as  the 
given  parallelogram.  Therefore, 
the  area  of  a  parallelogram  is  equal 
to  the  product  of  its  base  and  alti- 
tude, or  A  =  ah. 

188.  The  Triangle.  Since  the  line  AB  divides  the 
parallelogram  into  two  equal  triangles  with  the  same  base 
and  altitude  as  the  parallelogram,  the 
area  of  the  triangle  is  equal  to  one 
half  of  the  area  of  the  parallelogram. 
But  the  area  of  the  parallelogram  is 
equal  to  the  product  of  its  base  and 
altitude.  Therefore,  the  area  of  the  triauf/le  is  one  half 
tjie  product  of  its  base  and  altitude,  or  A  =  |  ab. 

Thus,  the  area  of  a  triangle  with  base  6  in.  and  altitude  5  in.  is  ^ 
of  6  X  5  sq.  in.  =  15  sq.  in. 

189.  The  Trapezoid.  The 
line  AB  divides  the  trapezoid 
into  two  triangles  whose  bases 
are  the  upper  and  lower 
bases  of  the  trapezoid  and  whose  common  altitude  is  the 


102 


MENSURATION 


altitude  of  the  trapezoid.  The  areas  of  the  triangles  are 
respectively  J  ab-^^  and  J  ab^,  and  since  the  area  of  the 
trapezoid  equals  the  sum  of  the  areas  of  the  triangles,, 
therefore  the  area  of  the  trapezoid  is  w(ib^-\-  ^  ab^  = 
1  a(b^  +  ^2)1  01'  ^^*^  ^'^^^(^  ^f  ^  trapezoid  is  equal  to  one  half 
of  the  product  of  its  altitude  and  the  sum  of  the  upper  and 
lower  bases,  or  A  =  ^  a(^b-^  +  ^2)- 

Thus,  the  area  of  a  trapezoid  whose  bases  are  20  ft.  and  17  ft.  and 
whose  altitude  is  6  ft.  is  1  of  6  x  (20  +  17)  sq.  ft.  =  111  sq.  ft. 

190.  The  Right  Triangle.  The  Hindu  mathematician 
Bhaskara  (born  1114  a.d.)  arranged  the  figure  so  that  the 
square  on  the  hjq^otenuse  contained  four  right  triangles, 
leaving  in  the  middle  a  small  square   whose  side  equals 


\     1 

2    \ 

5 

B 

■ 

4       /^ 
3 

the  difference  between  the  sides  of  the  right  triangle.  In 
a  second  figure  the  small  square  and  the  right  triangles 
were  arranged  in  a  different  Avay  so  as  to  make  up  the 
squares  on  the  two  sides.  Bhaskara's  proof  consisted 
simpl}^  in  drawing  the  figure  and  writing  the  one  word 
"Behold."  From  these  figures  it  is  evident  that  the  area 
of  the  square  constructed  on  the  hypotemise  will  equal  the 
sum  of  the  areas  of  the  squares  constructed  on  the  two  sides. 
In  general,  if  the  sides  of  the  right  triangle  are  a  and  b 
and  the  hypotenuse  is  c,  a^  -{-  b^  =  c^. 

Thus,  the  hypotenuse  of  the  right  triangle  whose  sides  are  5  and 
12  is  \/25T~lti  =  13. 


MENS  URA  TION  103 

This  theorem  is  known  by  the  name  of  the  Pythagorean  theorem, 
because  it  is  supposed  to  liave  been  first  proved  by  the  Greek  mathe- 
matician Pythagoras,  about  500  B.C. 

191.  If  either  side  and  the  hypotenuse  of  a  right  tri- 
ansfle  are  known,  the  other  side  can  be  found  from  the 
equation  a^  +  b'^  =  c^. 

Thus,  if  one  side  is  3  and  the  hypotenuse  is  5,  the  other  side  is 
V25  -9  =  4. 

EXERCISE   29 

1.  The  t^yo  sides  of  a  right  triangle  are  6  m.  and  8  in. 
Find  the  length  of  the  hypotenuse. 

2.  Find  the  area  of  an  isosceles  triangle,  if  the  equal 
sides  are  each  10  ft.  and  the  base  is  4  ft. 

3.  Find  the  area  of  an  isosceles  trapezoid,  if  the  bases 
are  10  ft.  and  18  ft.  and  the  equal  sides  are  8  ft. 

4.  What  is  the  area  in  hectares,  etc.,  of  a  field  in  the 
form  of  a  trapezoid  of  which  the  bases  are  475™  and 
580™  and  the  altitude  is  1270™? 

5.  Show  that  the   altitude   of  an   equilateral  triangle, 

each  of  whose  sides  is  a,  is  -VS. 

A 

6.  The  hypotenuse  of  a  right  triangle  with  equal  sides 
is  10  ft.     P'ind  the  length  of  the  two  equal  sides. 

7.  The  diagonal  of  a  square  field  is  80  rd.  How  many 
acres  does  the  field  contain  ? 

8.  Find  correct  to  centimeters  the  area  of  an  equilateral 
triangle  each  side  of  which  is  1™  in  length. 

192.  The  Circle.  If  the  circumference  (c)  and  the 
diameter  {cT)  of  a  number  of  circles  are  carefully  meas- 


1 04  MENS  URA  TION 


ured,  and   if  the   quotient  -   is  taken   in  each   case,  the 

ct 

quotients  will  be  found  to  liave  nearly  the  same  value. 
If  absolutely  correct  measurements  could  be  made,  the 
quotient  in  each  case  would  be  the  same  and  equal  to 
3.14159  +  ,  i.e.  the  ratio  of  the  circumference  of  a  circle 
to  its  diameter  is  the  same  for  all  circles.  The  ratio  is 
denoted  by  the  Greek  letter  tt  (pi).  The  value  of  tt 
found  in  geometry  is  3. 14159 +  .  In  common  practice  ir 
is  taken  as  3.1416.  The  value  of  tt  cannot  be  exactly 
expressed  by  any  number,  but  can  be  found  correct  to 
any  desired  number  of  decimal  places. 

193.    Since  -  =  tt  and  t?  =  2  r,  where  r  stands  for  the 
d  ^ 

radius  of  the  circle,   then   c  =  Trd  =  2  irr,  and  d  =  —   and 

TT 

r  =  TT— •  Hence,  if  the  radius,  diameter  or  circumference 
of  a  circle  is  known,  the  other  parts  can  be  found. 

Thus,   the   circumference   of   a   circle   whose   radius  is  10  in.   is 
2  X  3.1416  X  10  in.  =  62.832  in. 


EXERCISE   30 

1.  Find  the  circumference  of  a  circle  whose  diameter 
is  20  in. 

2.  Find  the  radius  of  a  circle  whose  circumference  is 
250  ft. 

3.  If  the  length  of  a  degree  of  the  earth's  meridian  is 
69.1  mi.,  what  is  the  diameter  of  the  earth  ? 

4.  If  the  radius  of  a  circle  is  8  in.,  w^hat  is  the  lengtli 
of  an  arc  of  15°  20'  ? 

5.  The  diameter  of  a  circle  is  10  ft.      How  many  de- 
grees are  there  in  an  arc  of  16  ft.  long  ? 


MENSURATION 


105 


194.  The  Area  of  a  Circle.  The  circle  may  be  divided 
into  a  number  of  equal  ligures  that  are  essentially  tri- 
angles. The  sum  of  the  bases  of 
these  triangles  is  the  circumference 
of  the  circle,  and  the  altitudes  are 
radii  of  the  circle.  Treating  these 
figures  as  triangles,  theii-  areas  will 
he  I  c  X  r.  Therefore,  since  (7  =  2  in\ 
A  =  \  oi  2  irr  X  r  =  7rr^.  It  is  proved  in  geometry  that  this 
result  is  exactly  correct. 

The  area  of  a  circle  whose  radius  is  5  ft.  is  3.1416  x  5  x  5  sq.  ft.  = 
78.51  sq.  ft. 

EXERCISE   31 

1.  Find  the  area  of  a  circle  whose  radius  is  10  in. 

2.  Find  the  area  of  a  circle  whose  circumference  is 
25  ft. 

3.  Find  the  radius  of  a  circle  whose  area  is  100  sq.  ft. 

4.  The  areas  of  two  circles  are  60  sq.  ft.  and  100  sq.  ft. 
Find  the  number  of  degrees  in  an  arc  of  the  first  that  is 
equal  in  length  to  an  arc  of  45°  in  the  second. 

5.  Find  the  side  of  a  square  that  is  equal  to  a  circle 
whose  circumference  is  50  in.  longer  than  its  diameter. 

195.  The  Volume  of  a  Rectangular  Parallelopiped.  If 
the   unit  of  measure  a  is  1  cu.  in., 

then   the  column  AB   is  4  cu.  in., 

and  the  whole   section  ABCD  will 

contain  3  of  these  columns,  or  3x4 

cu.  in.     Since  there  are  five  of  these  ;,     ■    j     vi  /  Jn 

sections   in   the   parallelopiped,    the 

entire  volume  ( T^)  is  5x3x4  cu. 

in.,  or  60  cu.  in.     Therefore,  the  vol- 


/ 

/■■///// 

V  /  /  AVi/ 

/  /'/  /  /bAAa 

//'. 

[ 





-M 

/ 

ywX 

/«  / 

106  MENS  URA  TION 

ume  of  a  rectangular  parallelopiped  is  equal  to  the  j^^oducts 
of  its  three  dimeyisions.  That  is,  the  number  of  cubic  units 
in  the  volume  is  equal  to  the  product  of  the  three  numbers 
that  represent  its  dimensions. 

196.  If  the  dimensions  of  the  rectangular  parallelopiped 
are  a,  b  and  c,  it  can  be  shown  in  the  same  way  that 
V=  abc.  Any  of  these  four  quantities,  V,  a,  5,  c,  can  be 
determined  when  the  other  three  are  known. 

Ex.     If  the  volume  of  a  rectangular  parallelopiped  is  36  cu.  in.  and 
o 
=  3.      .'.  3  in.  is  the  other  dimension. 


197.  If  the  dimensions  of  a  rectangular  paralleloj^iped 
are  equal,  the  figure  is  a  cube  and  the  volume  is  equal  to 
the  third  power  of  the  number  denoting  the  length  of  its 
edge  (a),  or  V=  a^.  For  this  reason  the  third  power  of  a 
number  is  called  its  cube. 

198.  It  is  proved  in  geometry  that  any  parallelopiped 
has  the  same  volume  as  a  rectangular  parallelopiped  with 
the  same  base  and  altitude. 

EXERCISE   32 

1.  Find  the  volume  of  a  cube  3  in.  on  an  edge. 

2.  Find  the  volume  of  a  rectangular  parallelopiped 
whose  edges  are  3*'"',  S'^™  and  11*'"\ 

3.  The  volume  of  a  rectangular  parallelopiped  is  100 
cu.  in.     The  area  of  one  end  is  20  sq.  in.     Find  the  length. 

4.  How  many  cubic  feet  of  air  are  there  in  a  room 
12  ft.  6  in.  long,  10  ft.  8  in.  wide  and  9  ft.  higli  ? 

5.  Find  the  weiglit  of  a  rectangular  block  of  stone  at 
135  lb.  per  cubic  foot,  if  the  length  of  tlie  block  is  9^-  ft. 
and  the  otlier  dimensions  are  2  ft.  and  5  ft. 


MENSURATION 


107 


6.  If  a  cubic  foot  of  water  weighs  1000  oz.,  find  the 
edge  of  a  cubical  tank  that  will  hold  2  T. 

7.  Show  why  the  statement  that  a  rectangular  parallelo- 
piped  is  equal  to  the  product  of  its  three  dimensions  is 
the  same  as  the  statement  tliat  its  volume  is  equal  to  the 
product  of  its  altitude  and  the  area  of  its  base. 


199.  The  Volume  of  a  Prism.  A  rectangular 
parallelopiped  can  be  divided  into  two  equal 
triangular  prisms  with  the  same  altitude  and 
half  the  base.  Hence,  the  volume  of  the 
prism  is  half  the  volume  of  the  parallelopiped. 
But  the  base  of  the  parallelopiped  is  twice 
the  base  of  tlie  prism,  therefore,  the  volume  of 
a  triangular  prism  is  equal  to  the  2^^'oduct  of 
its  altitude  and  the  area  of  its  base. 

200.  Since  any  prism  can  be  divided  into 
triangular  prisms,  as  in  the  figure,  it  follows 
that  the  volume  of  any  prism  is  equal  to  the 
product  of  its  altitude  and  the  area  of  its 
base. 

201.  The  Volume  of  a  Cylinder.  The 
cylinder  may  be  divided  into  a  number  of 
splids  that  are  essentially  prisms,  as  indi- 
cated in  the  figure.  The  sum  of  the  bases 
of  these  prisms  is  the  base  of  the  cylinder 
and  the  altitude  of  the  prisms  is  the  same 
as  the  altitude  of  the  cylinder.  There- 
fore, the  volume  of  a  cylinder  is  the  product 
of   its    altitude    and    the    area    of    its    base. 


a  xirr 


108  MENSURATION 


EXERCISE   33 


1.  Find  the  volume  of  a  prism  with  square  ends,  each 
side  measuring  1  ft.  8  in.,  and  the  height  being  12  ft. 

2.  Find  the  volume  of  a  prism  whose  ends  are  equilateral 
triangles,  each  side  measuring  11  in.  and  the  height  being 
20  in. 

y  =  aOC.        £ 

determined  ^^^®  volume  of  a  cylinder  if  the  diameter  of  its 

^       T.  .1  11-  i^iid  the  altitude  is  30  in. 
Ex.    If  th 

two  of  the  cmany  cubic  yards  of  earth  must  be  removed  in 

J^  =  3.     .vvell  45  ft.  deep  and  3  ft.  in  diameter  ? 
6x2  ^ 

3ic  foot  of  copper  is  to  be  drawn  into  a  wire  -^^ 

■    in  diameter.     Find  the  length  of  the  wire, 
are  equal,  1 

o.    Mow  many  revolutions  of  a  roller  3J  ft.  in  length  and 

2  ft.  in  diameter  will  be  required  in  rolling  a  lawn  |  of  an 

acre  in  extent. 

7.  Show  how  to  find  the  surface  of  a  cylinder  by  divid- 
ing it  into  figures  that  are  essentially  parallelograms. 
Show  how  to  find  the  surface  of  a  prism. 

202.  The  Volume  of  a  Pyramid.  Let  AB  be  a  cube  and 
F  the  middle  ]_)oint  of  the  cube,  then  by  connecting  F  with 
B,  (7,  B  and  E  a  pyramid  with  a  square 
base  is  formed.  It  is  evident  that  by 
drawing  lines  from  F  to  each  of  the  ver- 
tices, the  cube  will  consist  of  six  such 
pyramids.  Hence,  the  volume  of  the 
pyramid  is  \  of  the  volume  of  the  cube. 
The  volume  of  the  cube  is  the  product  of 
its  altitude  and  the  area  of  its  base  BCBE.  Therefore, 
the  volume  of  the  pyramid  is  1  of  the  product  of  the  alti- 
tude of  the  cube  and  the  area  of  its  base.     But  the  base 


MENSURATION  109 

of  the  pyramid  is  the  base  of  the  cube  and  its  altitude 
is  J-  of  the  altitude  of  the  cube,  hence^  the  volume  of  the 
pyramid  is  one  third  of  the  product  of  its  altitude  and  the 
area  of  its  base.  In  geometry  this  is  proved  true  of  any 
pyramid. 

Ex.  If  the  altitude  of  a  pyramid  is  45"^,  and  a  side  of  its  square 
base  is  60™,  its  volume  is  i  of  45  x  (60^)'"^  =  54000'"^ 

203.  The  Volume  of  a  Cone.  The  cone 
may  be  divided  into  a  number  of  equal  fig- 
ures that  are  essentially  pyramids  as  indi- 
cated in  the  figure.  The  sum  of  the  bases 
of  these  pyramids  is  the  base  of  the  cone, 
and  their  altitudes  are  the  same  as  the  alti- 
tude of  the  cone.  Therefore,  the  volume  of 
a  cone  is  equal  to  one  third  of  the  product  of 
its  altitude  and  the  area  of  its  base. 

Ex.  If  the  altitude  of  a  cone  is  10  ft.  and  the  radius  of  its  base 
4  ft.,  its  volume  is  i  of  10  x  3.1416  x  4^  cu.  ft.  =  167.55  cu.  ft. 

EXERCISE   34 

1.  Show  that  the  pyramid  with  a  square  base  can  be 
divided  into  two  equal  pyramids  with  triangular  bases  and 
the  same  altitude  as  the  original  pyramid;  and  hence  show 
how  any  pyramid  may  be  similarly  divided. 

2.  Find  the  volume  of  a  cone  if  the  diameter  of  the  base 
is  16  in.  and  the  altitude  is  12  in.  Find  the  volume  if 
the  diameter  of  the  base  is  16  in.  and  the  slant  height  is 
12  in. 

3.  Show  how  to  find  the  surface  of  a  cone  by  dividing 
it  into  figures  that  are  essentially  triangles.  Show  how  to 
find  the  surface  of  a  pyramid. 


110 


MENSURATION 


4.  Find  the  volume  of  a  pyramid  if  the  area  of  its  base 
is  4  sq.  ft.  and  its  altitude  is  2  ft.  Find  the  volume  if 
the  base  is  4  feet  square  and  the  slant  height  is  2  ft. 

5.  How  much  canvas  is  necessary  for  a  conical  tent  8  ft. 
higli,  if  the  diameter  of  the  base  is  8  ft.  ? 

6.  The  radius  of  a  cylinder  is  8  ft.  and  its  altitude  is  10  ft. 
Find  the  altitude  of  a  cone  with  the  same  base  and  volume. 


204.  The  Surface  of  a  Sphere.  The  surface  of  a  sphere 
is  proved  in  geometry  to  be  equal  to  the 
area  of  4  great  circles  oi-  4  Trr^,  r  being  the 
radius  of  the  sphere.  This  can  be  shown 
by  winding  a  firm  cord  to  coA-er  a  hemi- 
sphere and  a  great  circle  as  indicated  in  the 
figure.  It  will  be  found  that  twice  as  much 
cord  is  used  to  cover  the  hemisphere  as  the 
great  circle,  therefore,  to  cover  the  whole 
sphere  4  times  as  much  would  be  required. 

Ex.    A  sphere  with  a  radius  of  6  in.  has  a  surface  of  4  x  3.14:16  x 
6^  sq.  in.  =  452.39+  sq.  in. 


205.  The  Volume  of  a  Sphere.  The  sphere  may  be 
divided  into  a  number  of  fig- 
ures that  are  essentially  pyra- 
mids, as  indicated  in  the  figure. 
The  sum  of  tlie  bases  of  these 
pyramids  is  the.  surface  of  the 
sphere  and  the  altitude  of  each 

pyramid  is  its  radius.     Tlierefore,  tlie  volume  of  all  tliese 
pyramids  is  equal  to  J  r  x  4  7rr^  =  ^  irr^. 

Ex.    The  vohin>e  of  a  sphere  wliose  radius  is  3  in.  is  |  x  3.1416 
X  3^^  cu.  in.  =  113.1  cu.  in. 


MENS  URA  TION  111 

206.  Board  Measure.  In  measuiing  lmnl)er  the  board 
foot  is  used.  It  is  a  board  1  ft.  long,  1  ft.  wide  and  1  in. 
or  less  thick.  Lumber  more  than  1  in.  tliick  is  m(;asured 
by  the  number  of  square  feet  of  l)oards  1  in.  thick  to 
which  it  is  equal. 

Thus,  a  board  10  ft.  long,  1  ft.  wide  and  1},  in.  thick, 
contaius  15  board  feet. 

Lumber  is  usually  sold  by  the  1000  board  feet.  A  quo- 
tation of  !^17  per  jNI,  means  flT  per  1000  board  feet. 

EXERCISE  35 

1.  Find  the  cost  of  12  boards  16  ft.  long,  6  in.  wide, 
and  1  inch  thick  at  -f  18  per  M. 

2.  How  many  board  feet  are  there  in  a  stick  of  timber 
16  ft.  by  16  in.  by  10  in.  ? 

3.  How  much  is  a  stick  of  timber  15  ft.  by  2  ft.  by  1  ft. 
4  in.  worth  at  #22  per  M  ? 

4.  How  many  board  feet  are  used  in  laying  the  flooring 
of  a  two-story  house  32  ft.  by  20  ft.,  allowing  40  ft.  waste  ? 

5.  What  is  the  cost  of  25  21 -in.  planks  16  ft.  long  by 
1  ft.  wide  at  .$22.50  per  M? 

6.  What  is  the  cost  of  15  joists  12  ft.  by  10  in.  by  4  in. 
at  $23  per  M  ? 

207.  Wood  Measure.  The  unit  of  wood  measure  is  the 
cord.     The  cord  is  a  pile  of  wood  8  ft.  by  4  ft.  by  4  ft. 

A  pile  of  wood  1  ft.  b}^  4  ft.  by  4  ft.  is  called  a  cord 
foot. 

A  cord  of  stove  wood  is  8  ft.  long  by  4  ft.  high.  The 
length  of  stove  wood  is  usually  16  in. 


112  MENS  URA  TtON 

EXERCISE  36 

1.  Find  the  number  of  cords  of  wood  in  a  pile  32  ft.  by 
4  ft.  by  4  ft. 

2.  At  $5.75  per  cord,  how  much  will  a  pile  of  wood 
52  ft.  by  4  ft.  by  4  ft.  cost  ? 

3.  How  much  will  a  pile  of  stove  wood  94  ft.  long  4  ft. 
high  be  worth  at  $2.75  per  cord  ? 

208.  Carpeting.  A  yard  of  carpet  refers  to  the  running 
measurement,  regardless  of  the  width.  The  cheaper  grades 
of  carpet  are  usually  1  yd.  wide,  and  the  more  expensive, 
such  as  Brussels,  Wilton,  etc.,  are  J  of  a  yard  wide. 

In  carpeting,  it  is  usually  necessary  to  allow  for  some 
waste  in  matching  the  figures  in  patterns.  Dealers  count 
this  waste  in  their  charges.  In  computing  the  cost  of 
carpets,  dealers  charge  the  same  for  a  fractional  width  as 
for  a  whole  one. 

Carpets  may  often  be  laid  with  less  waste  one  way  of 
the  room  than  the  other;  hence,  it  is  sometimes  best  to 
compute  the  cost  with  the  strips  running  both  ways,  and 
by  comparison  determine  which  involves  the  smaller  waste. 

EXERCISE  37 

1.  How  many  yards  of  Brussels  carpet  |  of  a  yard  wide 
will  be  required  to  cover  the  floor  of  a  room  15  ft.  by  13  ft. 
6  in.,  tlie  waste  in  matching  being  4  in.  to  each  strip  ? 
Which  will  be  the  more  economical  way  to  lay  the  carpet  ? 

2.  How  much  will  it  cost  to  cover  the  same  room  with 
Brussels  carpet  if  a  border  |  of  a  yard  wide  is  used,  the 
carpet  and  border  being  SI. 25  per  yard,  and  the  waste 
being  4  in.  to  each  strip  of  carpet  and  -|  of  a  yard  of  border 
at  each  corner  ? 


MENSURATION  113 

3.  How  much  will  it  cost  to  cover  the  same  room  with 
ingi'ain  carpet  1  yd.  wide,  at  (-)7|  ct.  per  yard,  the  waste 
being  6  in.  to  eacli  strip  ? 

4.  At  $lA2}y  per  yard,  how  mucli  will  it  cost  to  carpet 
a  flight  of  stairs  of  14  steps,  each  step  being  8  in.  high 
and  11  in.  wide  ? 

5.  A  room  is  16  ft.  10  in.  by  14  ft.  9  in.  How  long 
must  the  strips  of  carpet  used  in  covering  the  floor  be  cut, 
if  tlie  pattern  is  14  in.  ?  (If  laid  lengthwise  of  the  room, 
the  length  of  15  patterns  must  be  used.)  Will  it  be 
clieaper  to  run  the  strips  lengthwise  or  across  the  room  ? 
If  the  room  is  covered  with  carpet  |  of  a  yard  wide  at 
11.35  per  yard,  how  much  will  it  cost? 

209.  Papering.  Wall  paper  is  sold  in  single  rolls  8  yd. 
long,  or  in  double  rolls  IG  yd.  long.  It  is  usually  18  in. 
wide. 

There  is  considerable  waste  in  cutting  and  matching 
paper.  AVhole  rolls  may  be  returned  to  the  dealer,  but 
part  of  a  roll  will  not  usually  be  taken  back. 

EXERCISE  38 

1.  How  many  rolls  of  paper  are  used  in  papering  a 
room  14  ft.  by  12  ft.  6  in.,  and  8  ft.  high  above  the  base- 
board, if  the  room  contains  2  windows  6  ft.  by  3  ft.  6  in. 
and  2  doors  7  ft.  by  4  ft.,  the  width  of  the  border  being 
16  in.,  and  6  in.  waste  being  allowed  to  each  strip  for 
matching  ? 

2.  How  much  will  it  cost  to  paper  the  above  room  if 
the  paper  is  11  ct.  per  roll,  the  border  being  18  in.  wide,  and 
the  paper  hanger  working  8  hr.  at  30  ct.  per  hour  ? 

ltman's  adv.  ar. — 8 


114  MENS  URA  TION 

3.  At  25  ct.  per  roll,  how  much  will  it  cost  to  paper  a 
room  18  ft.  square  and  9  ft.  high  above  the  baseboard, 
allowing  for  2  doors,  each  7  ft.  by  3  ft.  9  in.,  and  3  win- 
dows, each  6  ft.  by  3  ft.  4  in.,  the  border  being  18  in.  wide  ? 

210.  Painting  and  Plastering.  The  square  yard  is  the 
unit  of  painting  and  plastering. 

There  is  no  uniform  j^i'^ctice  as  to  allowances  to  be 
made  for  openings  made  by  windows,  doors,  etc.,  and  the 
baseboard.  To  avoid  complications,  a  definite  written 
contract  should  always  be  drawn  up. 

EXERCISE  39 

1.  How  much  will  it  cost  to  plaster  the  walls  and  ceiling 
of  a  room  15  ft.  by  13  ft.  6  in.,  and  9  ft.  high,  at  27J  ct. 
per  square  yard,  deducting  half  of  the  area  of  2  doors,  each 
7  ft.  by  31  ft.,  and  2  windows,  each  6  ft.  by  31  ft.  ? 

2.  How  much  will  it  cost  to  paint  the  walls  and  ceiling 
of  the  same  room  at  121  ct.  per  square  yard,  the  same 
allowance  being  made  for  openings  ? 

3.  At  20  ct.  per  square  yard,  how  much  will  it  cost  to 
paint  a  floor  18  ft.  by  16  ft.  6  in.  ? 

4.  Allowing  1  of  the  surface  of  the  sides  for  doors, 
windows  and  baseboard,  how  much  will  it  cost  to  plaster 
the  sides  and  ceiling  of  a  room  22  ft.  by  18  ft.  and  9^  ft. 
high,  at  221  ct.  per  square  yard  ? 

211.  Roofing  and  Flooring.  A  square  10  ft.  on  a  side, 
or  100  sq.  ft.,  is  the  unit  of  roofing  and  flooring. 

The  average  shingle  is  taken  to  be  16  in.  long  and  4  in. 
wide.     Shingles  are  usually  laid  about  4  in.  to  the  weather. 


ME^  S  URA  TION  115 

Allowing  for  waste,  Jibout  1000  sliinglt's  are  estimated  as 
needed  for  each  square,  but  if  the  shingles  are  good,  850 
to  900  are  suflieient.     Tliere  are  250  sliingles  in  a  bundle. 

EXERCISE  40 

1.  At  #8.60  per  square,  how  mucli  will  it  cost  to  sliingle 
a  roof  50  ft.  by  22^  ft.  on  each  side  ? 

2.  How  much  wdll  it  cost  to  lay  a  hard-wood  floor  in  a 
room  30  ft.  by  28  ft.,  if  the  labor,  nails,  etc.  cost  i$22.50, 
lumber  being  i28  per  M,  and  allowing  57  sq.  ft.  for  waste  ? 

3.  Allowing  900  shingles  to  the  square,  how  many 
bundles  will  be  required  to  shingle  a  roof  70  ft.  by  28  ft. 
on  each  side?  How  much  will  the  shingles  cost  at  $^3.75 
per  M  ? 

4.  At  i?  12.50  per  square,  how  much  will  the  slate  for  a 
roof  40  ft.  by  21:  ft.  on  each  side  cost  ? 

212.  Stonework  and  Masonry.  The  cubic  yard  or  the 
perch  is  the  unit  of  stonework. 

A  perch  of  stone  is  a  rectangular  solid  16J  ft.  by  1^  ft. 
by  1  ft.,  and  therefore  contains  24|  cu.  ft. 

A  common  brick  is  8  in.  by  4  in.  by  2  in.  Bricks  are 
usually  estimated  by  the  thousand,  sometimes  by  the  cubic 
f(5ot,  22  bricks  laid  in  mortar  being  taken  as  a  cubic  foot. 

There  is  no  uniformity  of  practice  in  making  allowances 
for  windows  and  other  openings.  There  should  be  a  defi- 
nite written  contract  with  the  builder  covering  this  point. 
The  corners,  however,  are  counted  twice  on  account  of  the 
extra  work  involved  in  building  them.  It  is  also  gener- 
ally considered  that  the  work  around  openings  is  more 
difficult,  so  that  allowance  is  frequently  made  here. 


116  MENS  URA  TION 


EXERCISE   14 


1.  If  60  ct.  per  cubic  yard  was  paid  for  excavating  a 
cellar  30  ft.  by  20  ft.  by  7  ft.,  and  $4.75  a  perch  was  paid 
for  building  the  four  stone  walls,  18  in.  thick  and  extend- 
ing 2  ft.  above  the  level  of  the  ground,  what  was  the  total 
cost? 

2.  How  many  bricks  will  be  used  in  building  the  walls 
of  a  flat-roofed  building  90  ft.  by  60  ft.  and  20  ft.  high,  if 
the  walls  are  18  in.  thick  and  500  cu.  ft.  are  allowed  for 
openings  ? 

3.  How  much  will  it  cost  to  build  the  walls  described 
in  Ex.  2,  if  the  bricks  are  $8.50  per  ]M,  and  the  mortar  and 
brick-laying  cost  $3.50  per  M  ? 

4.  How  many  perch  of  stone  will  be  needed  for  the 
walls  of  a  cellar  30  ft.  by  22|  ft.  and  9  ft.  deep  from  the 
top  of  the  wall,  the  wall  being  18  in.  thick  ?  How  many 
perch  will  be  needed  for  a  cross  Avail  of  the  same  thick- 
ness, allowing  for  half  of  a  door  7  ft.  by  4  ft.  ?  How  much 
will  the  stone  cost  at  -^4.50  a  perch  ? 

213.  Contents  of  Cisterns,  Tanks,  etc.  The  gallon  or  the 
barrel  is  the  unit  of  measure  for  cisterns,  tanks,  etc. 

The  liquid  gallon  contains  231  cu.  in.  and  the  barrel 
311  gal. 

EXERCISE   42 

1.  How  many  gallons  of  water  will  a  tank  10  ft.  long, 
3  ft.  wide  and  3  ft.  deep  contain  ?     How  many  barrels  ? 

2.  How  many  gallons  of  Avater  will  a  cistern  10  ft.  deep 
and  10  ft.  in  diameter  contain  ?     How  many  barrels  ? 

3.  How  many  barrels  will  a  cylindrical  tank  5  ft.  higli 
and  3  ft.  in  diameter  contain  ? 


MENSUBATION  117 

4.  How  many  barrels  of  oil  will  a  tank  40  ft.  long  and 
6  ft.  in  diameter  contain  ? 

5.  Show  tliat  to  find  the  approximate  num])er  of  gaHons 
in  a  cistern  it  is  necessary  only  to  multiply  the  number 
of  cubic  feet  by  7^  and  subtract  from  tlie  product  ^-J-^-  of 
the  product.  Apply  this  method  to  eacli  of  tlie  al)Ove 
exercises. 

6.  How  many  gallons  will  a  cask  contain,  the  bnng 
diameter  being  24  in.,  the  head  diameter  20  in.  and  the 
lengtli  34  in.  ? 

Suggestion .    The  average  or  mean  diameter  is '—^^ -*  =  22  in. 

214.  Measuring  Grain  in  the  Bin,  Corn  in  the  Crib,  etc. 
There  are  2150.42  cu.  in.  in  every  bushel,  stricken  measure, 
and  2747.71  cu.  in.  in  every  bushel,  heaped  measure. 


EXERCISE  43 

1.  How  many  bushels  of  wheat  does  a  bin  8  ft.  by  7  ft. 
by  6  ft.  contain  ? 

2.  Show  that  multiplying  by  0.8  will  give  the  approxi- 
mate number  of  stricken  bushels  in  any  number  of  cubic 
feet,  and  dividing  by  0.8  will  give  the  approximate  number 
of  cubic  feet  in  any  number  of  stricken  bushels. 

3.  Show  that  multiplying  by  0.63  Avill  give  the  approxi- 
mate number  of  heaped  bushels  in  any  number  of  cubic 
feet,  and  dividing  by  0.63  will  give  the  approximate  num- 
ber of  cubic  feet  in  any  number  of  heaped  bushels. 

4.  How  deep  must  a  bin  10  ft.  by  8  ft.  be  to  hold  500 
bushels  of  wheat  ? 


118  MEXS  URA  TION 

5.  A  farmer  builds  a  corncrib  20  ft.  long,  10  ft.  high, 
8  ft.  wide  at  the  bottom  and  12  ft.  wide  at  the  top.  How 
many  heaped  bushels  of  corn  in  the  ear  will  the  crib  hold 
when  level  full  ?  If  the  ridge  of  the  roof  is  3  ft.  above 
the  top  level,  how  many  bushels  will  the  crib  hold  when 
filled  to  the  ridge  ? 

Suggestion.     The  average  width  of  the  crib  is  -^ — '^ '-  —  10  ft. 

6.  How  many  stricken  bushels  of  shelled  corn  are  there 
in  the  above  crib  if  3  half  bushels  of  ears  make  one  bushel 
of  shelled  corn  ? 

215.  Measuring  Hay  in  the  Mow  or  Stack.  The  only 
correct  way  to  measure  hay  is  to  weigh  it.  Hovv^ever,  it 
is  sometimes  convenient  to  be  able  to  estimate  the  number 
of  tons  in  a  mow  or  stack.  The  results  of  such  estima- 
tions can  be  only  approximately  correct,  as  different  kinds 
of  hay  vary  in  weight.  In  well-settled  mows  or  stacks, 
as  nearly  as  can  be  estimated,  15  cu.  yd.  make  one  ton. 
When  hay  is  baled,  10  cu.  yd.  make  a  ton. 

EXERCISE   44 

1.  Approximately  how  many  tons  of  hay  are  there  in  a 
mow  40  ft.  by  22  ft.  and  15  ft.  deep  ? 

2.  Approximately  how  many  tons  of  hay  are  there  in  a 
circular  stack  21  ft.  high  and  averaging  80  ft.  in  circum- 
ference ? 

3.  Approximately  how  many  tons  of  hay  are  there  in  a 
rick  averaging  35  ft.  long,  15  ft.  wide  and  20  ft.  high  ? 

216.  Land  Measure.    The  unit  of  land  measure  is  the  acre. 
In  the  Eastern  states,  the  land  was  divided,  as  convenient, 

when  settled,  and  the  description  of  tracts  of  land  refer  to 


MEN  SUE  A  TION 


119 


such  natural  objects  as  iiear-l)y  bowlders,  ponds,  estal)lished 
roads,  etc.  l>ut  all  states  whose  lands  have  been  surveyed 
since  1802  are  divided  by  a  system  of  meridians  and  paral- 
lels into  townships  G  miles  square.  Each  township  con- 
tains 36  scpiare  miles  or  sections.  Each  section  contains 
2  half  sections  and  4  quarter  sections. 

Public  lands  arc  located  with  i-cfcrence  to  a  nortli  and 
south  line  called  the  principal  meridian  and  an  east  and 
west  line  called  the  base  line.  The  nortli  and  south  rows 
of   townships    are  called  ranges  ^ 

and    these    rows    are    numbered 
from     the    principal     meridian.      ~ 
The    townships    are    numbered     — 

from  the  base  line.     A  township     

is    therefore    designated    by   its  ^ 
number  and   the   number   of  its     — 


Thus,  A  is  township  4  N".,  Range  3, 
W.     What  is  5?     J/?     S'i 


'  The  36  sections  of  a  township  are  numbered  as  in  the 
following  diagram.  The  corners  of  all  sections  are  per- 
manently marked  by  stones,  or  otherwise. 

A  SECTION 


A  TOWNSHIP 

6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

10 

15 

14 

13 

19 

20 

21 

■^2 

23 

21 

30 

29 

28 

27 

20 

25 

31 

32 

33 

31 

35 

36 

N.  i  Section 
(320  A.) 

S.W.  1 
(160  A.) 

W.i 

of 
S.E.i 

'SO  A 

S.E.i 

The  divisions  of  sections  into  half  sections,  quarter  sec- 
tions, etc.,  are  shown  in  the  diagram. 

Thus,  the  X.  E.  \  of  N.  E.  \,  section  6,  means  the  northeast  quarter 
of  the  northeast  quarter  of  section  6. 


120  MENSURATION 


EXERCISE   45 


1.  How  many  acres  are  there  in  a  section  ?  In  the 
S.  W.  \  of  S.  W.  1,  section  16  ?  In  S.  |  of  N.  E.  \,  section 
36  ?     Locate  these  sections. 

2.  What  will  be  the  cost  of  a  quarter  section  of  land  at 
$  bb  an  acre  ? 

3.  How  many  rods  of  fence  are  necessary  to  inclose  a 
quarter  section  ? 

4.  How  many  acres  are  there  in  a  township  ? 

5.  The  sections  of  a  township  are  separated  and  the 
township  is  separated  from  adjacent  townships  by  a  road 
45  ft.  wide,  the  section  lines  being  in  the  middle  of  the  road. 
How  many  acres  are  there  in  the  roads  of  the  township  ? 

EXERCISE  46 

1.  The  side  of  a  square  is  100  ft.  Find  the  length  of 
a  diagonal. 

2.  One  side  of  a  right-angled  triangle  is  16  yd.  and 
the  other  side  is  J  of  the  hypotenuse  ;  what  is  the  length 
of  the  hypotenuse  ? 

3.  Find  the  volume  of  a  pyramid  Avhose  base  and  faces 
are  all  equilateral  triangles  with  sides  10  in.  long. 

4.  The  largest  pyramid  in  the  world  has  a  square  base 
with  sides  764  ft.  Its  four  faces  are  equilateral  triangles. 
Find  the  number  of  acres  covered  by  its  base,  the  number 
of  square  yards  in  its  four  faces,  and  the  height  of  the 
pyramid. 

5.  A  cistern  22  ft.  long,  10  ft.  wide  and  8  ft.  deep  is 
to  be  filled  with  water  from  a  well  8  ft.  in  diameter  and 
40  ft.  deep.  If  no  water  flows  into  the  well  while  filling 
the  cistern,  find  how  far  the  Avater  in  the  well  is  lowered. 


MEN  SURA  TION  121 

6.  Two  persons  start  from  the  same  place  at  the  same 
time.  One  walks  clue  east  at  the  rate  of  3  mi.  an  hour, 
and  the  other  due  soutli  at  the  rate  of  3i  mi.  an  hour.  In 
how  many  hours  Avill  they  be  30  mi.  apart  ? 

7.  What  is  the  circumference  of  the  earth  if  its 
diameter  is  7916  mi.  ? 

8.  Air  being  0.00129206  as  heavy  as  water,  find  in 
kilograms  the  weight  of  tlie  air  in  a  room  23""  long,  16"' 
wide  and  10'"  high. 

9.  A  rectangular  sheet  of  tin  of  uniform  tliickness  is 
SS'^'"  wide  and  2.7'"  long,  and  weighs  356^.  Find  its  thick- 
ness if  tin  is  7.3  times  as  heavy  as  water. 

10.  A  plate  of  iron  weighs  277. 54^^,  and  is  137*^'"  long, 
643"""  wide,  43.1'"'"  thick.  How  much  heavier  than  watei' 
is  iron  ? 

11.  A  tank  is  2'"  long,  5^^'"  wide  and  8*'"  deep.  How 
many  liters  of  water  will  it  contain,  and  how  much  will 
the  water  weigh  ? 

12.  Sulphuric  acid  is  1.84  times  as  heavy  as  water. 
How  many  kilograms  will  a  tank  hold  that  is  2'"  long,  75^"' 
wide  and  50*^'"  deep  ? 

13.  A  block  of  marble  is  2  ft.  long,  10  in.   wide  and 

8  in.  thick.     What  is  the  edge  of  a  cubical  block  of  equal 
volume  ? 

14.  If  1  T.  of  hard  coal  occupies  a  space  of  36  cu.  ft., 
how  many  tons  will  a  bin  10  ft.   long,  7|   ft.  wide   and 

9  ft.  deep  liold  ? 

15.  How  much  space  will  a  car  load  of  hard  coal  con- 
sistiug  of  38  T.  14  cwt.  75  lb.  occupy,  if  one  ton  occupies 
36  cu.  ft.  ? 

16.  How  long  must  a  bin  20  ft.  wide  and  20  ft.  deep  be 
to  hold  the  above  car  load  of  coal  ? 


122  MENS  URA  TION 

17.  Find  correct  to  0.001  the  diagonal  of  a  square  whose 
side  is  10  in.,  and  the  diagonal  of  a  cube  whose  edge  is 
10  in. 

18.  What  will  be  the  expense  of  painting  the  walls  and 
ceiling  of  a  room  whose  height  is  10  ft.  4  in.,  length  16  ft. 
6  in.  and  width  12  ft.  3  in.,  at  15  ct.  per  square  yard  ? 

19.  At  11  ct.  per  square  foot,  how  much  will  it  cost  to 
make  a  cement  walk  5  ft.  wide  around  a  school  yard  in 
the  shape  of  a  rectangle,  18  rd.  by  26  rd.  ? 

20.  Two  corridors  of  a  public  building  intersect  at  right 
angles  near  the  center  of  the  building.  If  the  corridors 
are  160  ft.  and  140  ft.  long  respectively,  and  20  ft.  wide, 
how  much  will  it  cost  to  cover  them  with  a  hard-wood 
floor  at  $  24  per  thousand  feet  ?    . 

21.  At  $18  per  M,  how  much  will  it  cost  to  cover  the 
floor  of  a  barn  30  ft.  long  and  20  ft.  wide  with  2-inch 
planks  ? 

22.  How  much  will  it  cost  to  fence  the  school  yard 
mentioned  in  Ex.  19,  with  1-inch  boards,  6  in.  wide,  at 
$11. bO  per  M;  the  fence  to  be  4  boards  high  and  built 
2  ft.  inside  the  walk  ? 

23.  How  many  board  feet  are  there  in  150  rafters,  14  ft. 
long,  4  in.  wide  and  2  in.  thick  ? 

24.  How  many  bunches  of  shingles  will  be  required  to 
shingle  a  barn  with  a  roof  60  ft.  long  and  rafters  18  ft. 
long,  the  shingles  being  laid  4  in.  to  the  weatlier  with 
a  double  row  at  the  bottom  ? 

25.  What  is  the  value  of  a  log  that  will  cut  36  1-inch 
boards,  each  16  ft.  long  and  12  in.  wide  at  1|  ct.  per 
square  foot  ? 


MENS  I  'JiA  TION  123 

26.  How  many  board  feet  are  there  in  a  stick  of  timber 
18J  ft.  long,  lb  in.  wide  and  12  in.  thick  ? 

27.  How  many  bricks  will  be  used  in  l)uilding  the  walls 
of  a  building  120  ft.  long,  60  ft.  wide  and  45  ft.  higli, 
outside  measurement,  if  tlie  walls  are  18  in.  thick  and  no 
allowance  is  made  for  doors  and  windows  ? 

28.  How  many  centimeters  of  lead  are  there  in  a  piece 
of  lead  pipe  1'"  long,  the  outer  diameter  being  5*"',  and  the 
thickness  of  the  lead  being  10"""  ? 

29.  A  race  track  30  ft.  wide  with  semicircular  ends  is 
constructed  in  a  field  1050  ft.  by  400  ft.  Find  the  inside 
and  outside  lenofths  of  the  track.  Also  find  the  area  of 
the  track  and  the  area  of  the  field  inside  the  track. 

30.  Find  the  volume  and  convex  surface  of  a  cone,  the 
diameter  of  the  base  being  16  in.  and  the  altitude  18  in. 

31.  Find  the  volume  and  surface  of  a  sphere  whose 
diameter  is  6  in. 

32.  Find  the  least  possible  loss  of  material  in  cutting  a 
cube  out  of  a  sphere  of  wood  9  in.  in  diameter. 

33.  Find  the  least  possible  loss  of  material  in  cutting  a 
spliere  out  of  a  cubical  block  of  wood  with  edges  9  in.  long. 

34.  Find  the  cost  of  making  a  road  200  yd.  in  length 
and  24  ft.  wide ;  the  soil  being  first  excavated  to  the  depth 
of  14  in.,  at  a  cost  of  20  ct.  per  cubic  yard;  crushed  stone 
being  then  put  in  8  in.  deep  at  a  cost  of  40  ct.  per  cubic 
yard,  and  gravel  placed  on  top  6  in.  thick  at  a  cost  of 
45  ct.  per  cubic  yard. 

35.  A  map  of  Kansas  is  made  on  a  scale  of  1  in.  to 
100  mi.  The  map  measures  4  in.  by  2  in.  Find  the 
area  of  the  state. 


GRAPHICAL   REPRESENTATIONS 

217.  Graphical  methods  of  representing  relations  be- 
tween different  measurements  are  so  extensively  used  in 
many  lines  of  work  ^hat  it  seems  best  to  give  a  brief  treat- 
ment of  the  subject  here.  Such  graphical  representations 
as  are  given  in  the  following  exercises  show  relations  pic- 
torially  in  a  much  clearer  manner  than  can  be  shown  by  a 
mere  statement  of  figures. 

Ex.  1.  Explain  graphically  the  relation  between  an  inch 
and  a  centimeter.     The  two  lines  drawn 

accurately  to  scale  represent  graphically     LlLL 

the  relation  between  the  inch  and  the  i  cm. 

centimeter. 

Ex.  2.  Draw  a  line  1.5  in.  long  and  find  the  number  of 
centimeters  in  it. 

Ex.  3.    Explain  graphically  the  relation  1  lb- 

between    the    pound   and    the    kilogram, 
given  l«^s=2.2  lb. 

Ex.  4.  Explain  graphically  the  relation  between  a  pint 
and  a  liter,  given  1^  =  1.76  pt. 

Ex.  5.  Erom  a  diagram  find  (a)  the  number  of  centi- 
meters in  4  in.,  (h)  the  number  of  liters  in  a  gallon,  (^c)  the 
number  of  pounds  in  5^^. 

Ex.  6.  The  values  of  manufactures  produced  in  the 
United  States,  Germany,  France  and  Great  Britain  in 
18G0    were    i^  1907000000,     $1995000000,    $2092000000, 

124 


IT^'g  =  2.2  lb 


GRAPHICAL   liEPEESENTA  TIONS 


125 


$2808000000  I'L'spectively,  and  in  181)4  they  were 
19498000000,  13357000000,  ^2900000000,  14263000000 
respectively. 

These  facts  may  be  represented  grapliically  as  follows: 


/U.S. 
Q\  Germ. 
France 
Gt.Brit.l 


MILLIONS  OF  DOLLARS 
1907 

1995 
2092 
2808 


oil  Germ. 


"] 


France 
Gt.Brit. 


19498 
3357 
2900 
4263 


These  measurements,  drawn  accurately  to  a  scale,  show  at  a  glance 
the  comparative  grow^th  in  manufactures  produced  in  the  different 
countries  mentioned  from  1860  to  1894. 


Fx.  7.  The  areas  of  England  and  Michigan  are  50839 
and  58915  square  miles  respectively.  The  populations  are 
approximately  31000000  and  2421000.  Represent  graph- 
ically the  comparative  sizes  and  the  comparative  density 
in  population  of  the  two. 

The  square  roots  of  the  numbers  representing  the  areas  correct  to 
units'  place  are  225  and  243  respectively.     The  ratio  between  these 


England 


Michigan 


two   numbers   reduces    to    5    to    5.4.      If    some    convenient   unit   of 
measure  be  taken,  and  squares  be  constructed  witli  sides  equal  to  5 


126  GRAPHICAL   REPRESENTATIONS 

to  5.4  of  these  units,  these  squares  will  represent  graphically  the 
comparative  areas.  The  comparative  density  in  population  will  be 
represented  by  the  number  of  dots  that  appear  in  each  square,  it  being 
assumed  that  a  dot  represents  100000  in  population.  There  will  then 
be  310  dots  in  the  square  representing  England  and  24  in  the  square 
representing  Michigan. 

^x.  8.  On  a  certain  day  between  6  A.M.  and  7  P.M. 
the  thermometer  registers  as  follows  :  6  A.M.,  20°  ;  7  A.M., 
22.5°;  8  a.m.,  27°;  9  a.m.,  35°;  10  a.m.,  42.5° ;  11  a.m., 
48°;  12  M.,  52°;  1  p.m.,  5b° ;  2  p.m.,  60°;  3  p.m.,  62°; 
4  p.m.,  60°;  5  p.m.,  50°;  6  p.m.,  42°;  7  p.m.,  35°.  Illus- 
trate graphically  this  variation  in  temperature. 

Draw  two  straight  lines  perpendicular  to  each  other.  Measure  off 
on  the  horizontal  line  OX  equal  spaces,  each  representing  1  hr.,  and 
on  the  perpendicular  line  OF  equal  spaces,  each  one  representing  10°. 
The  temperature  at  0  a.m.  is  shown  at  0;  at  7  a.m.  at  A,  a  distance 

TEMPERATURE 

70°Ly 

60" 

50" 

40' 

30' 


A.  M.  M.  P.  M. 

I \ \ I \ I \ 


06         7         8         910        1112         12         3^5  67 


X 


of  1  unit  along  OX  and  |  of  a  unit  above  OX  parallel  to  OY;  at  8  a.m. 
at  B,  a  distance  of  2  units  along  OX  and  .7  of  a  unit  above  OX  parallel 
to  OY.  In  the  same  way  points  may  be  located  showing  the  tempera- 
ture at  each  hour.  A  continuous  curve  drawn  through  these  points 
is  the  temperature  curve  for  the  day  from  0  a.m.  to  7  p.m.  This  curve 
shows  at  a  glance  the  variation  in  temperature  between  the  hours 
given. 


GRAPHICAL   REPRESENTATIONS 


127 


6  hr.  Y 


Ux.  9.  Two  trains  leave  a  certain  place  traveling  in  tlie 
same  direction,  one  at  the  rate  of  20  mi.  an  hour,  and  the 
other  at  the  rate  of  40  mi.  an  hour.  If  the  second  train 
leaves  3  hr.  after  the  first,  when  and  where  will  it  pass  the 
first? 

Let  each  space  along  OX  represent  20  mi.,  and  each  space  along 
OY  represent  1  hr.  At  the  end  of  the  first  hour  the  first  train  is  at  A  ; 
at  the  end  of  the  second 

hour  at  B,;  and  at  the    ^,,„  v  P^ 

end  of  the  sixth  hour 
at  P.  At  the  end  of 
the  fourth  hour  the  sec- 
ond train,  which  starts 
from  0',  3  spaces  above 
O,  since  it  starts  3  hr. 
later,  is  at  .4';  at  the 
end  of  the  fifth  hour  at 
B' ;  and  at  the  end  of 
the  sixth  hour  at  P. 
The  point  P,  where  the 
line  OP  and  O'P  cross, 
is  the  place  where  the 
second  train  overtakes  the  first.  If  from  P  perpendiculars  PX  and 
PF  are  dropped  upon  OX  and  OY,  then  the  distances  OX  and  OY 
will  represent  the  space  traveled  and  the  time  that  has  elapsed  since 
the  starting  of  the  first  train  till  the  second  one  overtakes  it.  OX 
contains  6  distance  spaces,  and  represents  120  mi.,  while  OF  contains 
6  time  spaces,  and  represents  6  hr. 


EXERCISE   47 

For  convenience  in  constructing  the  graphical  repre- 
sentations required  in  the  following  exercises,  the  student 
should  provide  himself  with  paper  ruled  in  small  squares. 

1.  Illustrate  graphically  the  comparative  areas  and  the 
comparative  density  in  population  in  the  following  cases  : 


128 


GBAPIIICAL   BEPEESENTA  TIONS 


Area 


POFL  LATIoN 


Q,)   Alaska 

Greenland 

(J>)   Mexico 

Texas     

(c)  United  States  (including  foreign 
possessions) 

British  Empire 


590884 
8;:I7837 
767258 
265780 

3806279 
11391036 


63592 

12000 

13606000 

3048710 

84907156 
383165494 


2.  On  Jan.  1,  1904,  the  thermometer  registered  the 
temperature  at  1  a.m.,  and  at  each  succeeding  hour  till 
midnight,  at  Ypsilanti,  Michigan  and  Havana,  Cuba, 
respectively  as  follows : 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

N. 

23° 

24° 

24° 

24° 

22° 

22° 

21° 

19° 

18° 

20° 

21° 

22° 

64° 

64° 

63° 

63° 

63° 

63° 

62° 

64° 

67° 

68° 

70° 

72° 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

Mt. 

22° 

22° 

22° 

21° 

17° 

16° 

16° 

15° 

13° 

12° 

12° 

12^^ 

73° 

73° 

73° 

73° 

72° 

70° 

69° 

68° 

67° 

66° 

65° 

Qb" 

Illustrate  each  graphically. 

3.  The  mean  temperature  for  January  (average  for  the 
31  da.  of  the  month)  for  the  same  hours  and  places  as  in 
Ex.  2  was  as  follows : 

1234567  89         10        UN. 

14.2°  14.3°  14.2°  14.2°  14.1°  14.3°  14.2°  14.5°  15.3°  17.1°  18.1°  19.7° 
67.2°  67.0°  66.7°  66.4°  66.0°  65.6°  65.4°  66.6°  68.9°  71.0°  73.1°  74.0° 

12  3  4  5  6  7  8  9         10        11       Mt. 

20.4°  20.7°  20.6°  19.9°  18.8°  18.1°  17.5°  16.5°  16.0°  15.4°  14.5°  16.6° 
74.7°  74.9°  75.0°  74.7°  74.2°  72.9°  71.6°  70.6°  69.8°  69.0°  68.4°  67.9'^ 


Illustrate  graphically. 


GRAPHICAL   n EPRESENTATIONS  129 

4.  Illustrate  graphically,  as  in  Ex.  9,  the  point  where 
and  time  at  which  the  two  trains  given  in  the  annexed 
time-table  pass  each  other. 


GoixG  East 

Going  West 

A.M. 

Miles 

Miles 

A.M. 

10.00 

284 

Detroit 

0 

12.35 

8.54 

247 

Ann  Arbor 

37 

1.25 

8.00 
7.50 

214 

(  Lv.  -,     ,          Ar.  ) 
i    .      Jackson  _      r 
I  Ar.                   Lv.  ) 

76 

2.20 
2.25 

6.10 

164 

Battle  Creek 

121 

3.30 

4.55 

141 

Kalamazoo 

144 

4.10 

3.25 
3.15 

93 

(Lv.  ^,.,      Ar.  ) 

]   .      Niles  ^      >■ 
I  Ar.             Lv.  ) 

192 

5.28 
5.33 

1.55 

56 

Michigan  City 

228 

6.32 

12.40 

13 

Kensington 

271 

7.30 

12.00 

0 

Chicago 

284 

8.00 

night 

A.M. 

5.  A  cyclist  starts  at  7  a.m.  from  a  town  and  rides 
2  hr.  at  the  rate  of  10  mi.  an  hour.  He  rests  1  hr.  and 
then  returns  at  the  rate  of  9  mi.  an  hour.  A  second 
cyclist  leaves  the  same  place  at  8  A.M.  and  rides  at  the 
rate  of  6  mi.  an  hour.     When  and  where  will  they  meet  ? 

6.  Two  cyclists  start  from  the  same  place  at  the  same 
time.  The  first  rides  for  2  hr.  at  the  rate  of  9  mi.  an 
hour,  rests  15  min.,  and  then  continues  at  6  mi.  an  hour. 
The  second  one  rides  without  stopping  at  the  rate  of  7  mi. 
an  hour.     Where  will  the  sefcond  cyclist  overtake  the  first  ? 

7.  The  average  yield  of  wheat  per  acre  in  the  United 
States  for  the  years  from  1893  to  1903  in  bushels  was  as 
follows:  11.4,  13.2,  13.7,  12.4,  13.4,  15.3,  12.3,  12.3,15.0, 
14.5, 12.9.  The  highest  Chicago  cash  price  per  bushel  for 
the  same  years  given  in  cents  was:  64.5,  63f,  64|,  93i, 
109,  70,  691  755^  791  773^  87.  Illustrate  grapliically, 
putting  the  tw^o  curves  in  one  figure. 

lyman's  adv.  ar.  — 9 


130  GRAPHICAL   REPRESENTATIONS 

8.  The  average  yield  of  corn  per  acre  in  the  United 
States  for  the  years  from  1893  to  1903  in  bushels  was  as 
follows:  22.5,  19.4,  26.2,  28.2,  23.8,  21.8,  25.3,  25.3,  16.7, 
26.8,  26.5.  The  highest  Chicago  cash  price  per  bushel 
for  the  same  years  given  in  cents  was :  36 J,  17^,  26|,  23|, 
271  38,  311  401,  671  57^,  433.  Illustrate  graphically, 
putting  the  two  curves  in  one  figure. 

9.  The  average  summer  daily  temperature  in  Paris  at 
the  foot  and  top  of  the  Eiffel  tower  in  1900  was  as 
follows : 

2  4         6         8         10        N.        2         4         6  8         10       Mt. 

57.2°  55.4°  58.1°  63.5°  67.8°  69.8°  70.1°  69.8°  68°      62.1°  60.7°  58.9° 
57.4°  55.7°  57.2°  58.1°  60.8°  63.5°  63.9°  64°      64.4°  61.2°  60.7°  59.1'' 

Illustrate  graphically,  putting  the  two  curves  in  one 
figure. 


RATIO   AND   PROPORTION 

218.  The  ratio  of  one  number  to  another  of  the  same 
kind  is  tlieir  quotient.  The  former  number  is  called  the 
antecedent,  and  the  hitter  the  consequent.  The  terms  of 
the  ratio  therefore  bear  the  same  rehilion  to  each  other  as 
the  terms  of  a  fraction.     Thus,  the  ratio  of  a  to  h  may  be 

written  a  :  h  (read  the  ratio  of  a  to  i),  -  or  a^h.  The 
forms  a  :  6,  and  -,  are  generally  used.  The  ratio  of  3  ft. 
to  5  ft.  is  3  :  5.     This  may  also  be  expressed  by  |  or  0.6. 

219.  The  ratio  is  always  an  abstract  number,  since  it  is 
the  relation  of  one  number  to  another  of  the  same  kind. 
There  can  be  no  ratio  between  5  hr.  and  %  10,  nor  between 
7  lb.  and  6  ft.  But  there  can  be  a  ratio  between  3  ft.  and 
6  in.,  since  the  quantities  are  of  the  same  kind.  Both 
terms  must,  however,  be  reduced  to  the  same  unit.  Thus, 
3  ft.  =  36  in.,  and  36  in. :  6  in.  =  ^{-  =  6. 

The  ratio  -   is  called  the  inverse  or  reciprocal  of  the 

.     a  ^ 

ratio  -• 

EXERCISE    48 

1.  How  is  the  value  of  a  ratio  affected  by  multiplying 
or  dividing  both  terms  by  the  same  number  ? 

2.  How  is  the  value  affected  by  multiplying  or 
dividing  the  antecedent  ?  by  multiplying  or  dividing 
the  consequent? 

131 


132  BATIO  AND  PROPORTION 

Express  the  ratio  of  : 

3.  100  to  25.  7.  115  to  50  cents. 

4.  16|  to  100.  8.  71  to  371. 

5.  331  to  100.  9.  I  to  16|. 

6.  2 "™  4  «^"  to  50  ^'".  10.  121  to  100. 

11.  14  lir.  30  mill.  3  sec.  to  a  day. 

12.  2  mo.  10  da.  to  a  year. 

13.  What  number  has  to  10  the  ratio  2  ?  to  5  the 
ratio  0.3? 

14.  li  X  :  S  =  5,  find  x. 

15.  li  X  :  1  =  2,  find  x. 

16.  Which  ratio  is  the  greater,  -f^  or  -f^  ^  if  ^^^  if  ^ 

17.  The  ratio  of  the  circumference  of  a  circle  to  its 
diameter  being  3.1416,  find  the  diameter  of  a  circle  whose 
circumference  is  125  ft.  correct  to  inches. 

18.  A  map  is  drawn  on  the  scale  of  1  in.  to  75  mi.  In 
what  ratio  are  the  lengths  diminished  ?  In  what  ratio  is 
the  area  diminished  ? 

19.  Two  rooms  are  14  ft.  long,  12  ft.  wide,  and  12  ft. 
long,  10  ft.  wide  respectively.  What  is  the  ratio  of  the 
cost  of  carpeting  them  ? 

20.  What  is  the  ratio  of  a  square  field  20  rd.  on  a  side 
to  one  25  rd.  on  a  side  ? 

21.  What  is  the  ratio  of  the  circumferences  of  two  cir- 
cles whose  diameters  are  2  in.  and  4  in.  ?  of  two  circum- 
ferences whose  diameters  are  5  in.  and  7  in.  ?  of  two 
circumferences  whose  diameters  are  d  and  c?'?  Hence  in 
general  the  ratio  of  two  circumferences  is  equal  to  wliat  ? 


RATIO  AND  PROPORTION  133 

22.  What  is  the  ratio  of  the  areas  of  two  circles  whose 
radii  are  3  in.  and.  5  in.  ?  of  the  areas  of  two  circles 
whose  radii  are  4  in.  and  6  in.?  of  the  areas  of  two  cir- 
cles whose  radii  are  r  and  r'  ?  Hence  in  general  the  ratio 
of  the  areas  of  two  circles  is  equal  to  what  ? 

23.  What  is  the  ratio  of  the  volumes  of  two  spheres 
whose  radii  are  2  in.  and  3  in.?  of  the  volumes  of  two 
spheres  whose  radii  are  5  in.  and  6  in.?  of  the  volumes 
of  two  spheres  whose  radii  are  r  and  r'  ?  Hence  in  general 
the  ratio  of  the  volumes  of  two  spheres  is  equal  to  what  ? 

220.  Specific  Gravity.  The  specific  gravity  of  a  sub- 
stance is  tlie  ratio  of  its  weight  to  the  weight  of  an  equal 
volume  of  some  other  substance  taken  as  a  standard. 

221.  Distilled  water  at  its  maximum  density,  4°  C,  is 
the  standard  of  specific  gravity  for  solids  and  liquids. 

222.  Since  I""''  of  water  weighs  1  gram,  the  same  num- 
ber that  expresses  the  weight  of  any  substance  in  grams 
will  also  express  its  specific  gravity.  Thus,  1^"'^  of  water 
weighs  1^;  hence,  1  is  the  specific  gravity  of  water.  1"^^ 
of  lead  weighs  11.35^;  hence,  this  being  11.35  heavier 
than  an  equal  volume  of  water,  the  specific  gravity  of 
lead  is  11.35. 

Specific  Gravities  of  Substances 

Copper  .     .    8.92  Tin 7.29  Sea  Water  .     .  1.026 

Iron  (cast)     7.21  Anthracite  Coal  1.30  Sulphuric  Acid  1.811 

Gold.     .     .19.26  Cork     .     .     .     .0.24  Milk   .     .     .     .1.032 

Lead      .     .  11.30  Pine      ....  0.65  Alcohol  .     .     .  0.84 

Platinum  .  21.50  Oak 0.845  Ice 0.92 

Mercury     .  13.596  Beech   ....  0.852  Rock  Salt    .     .  2.257 

1  cu.  ft.  of  water  weighs  about  1000  oz.,  or  62.5  lb. 


134  RATIO  AND  PROPORTION 

Ex.  1.    A  mass  of  cast  iron  weighs  3500  lb.     How  many 
cubic  feet  does  it  contain  ? 

Since  1  cu.  ft.  of  water  weighs  62.5  lb.,  1  cu.  ft.  of  iron  weiglis 
7.21  X  62..5  lb. 
3500 


'.21  X  62.5 


7.77,  the  number  of  cubic  feet. 


Ex.  2.  In  France  wood  is  sold  by  weight.  How  much 
does  1  stere  of  beech  wood  weigh,  allowing  ^  for  space 
not  filled  ? 

Since  1  ""^  of  water  weighs  1000  ^s,  1  stere  of  beech  wood  weighs 
0.852  X  1000  Kg  _  i  of  0.852  x  1000  ^g  =  568  Kg. 

EXERCISE    49 

1.  What  is  the  ratio  of  the  weight  of  1  stere  to  1  cord 
of  oak  wood,  allowing  J  for  waste  space  ? 

2.  Allowing  ^  for  waste  space,  how  many  tons  of  coal 
will  a  bin  9  ft.  long,  8  ft.  wide  and  8  ft.  deep  hold  ? 

3.  What  is  the  weight  of  a  cubic  decimeter  of  each  of 
the  substances  in  the  above  table  ?    of  a  cubic  foot  ? 

4.  A  flask  will  hold  6  oz.  of  w^ater.  How  much  alcohol 
will  it  hold  ?   how  much  mercury  ? 

5.  To  what  depth  will  a  cubic  foot  of  cork  sink  in  sea 
water  ?   in  alcohol  ? 

6.  How  much  does  a  piece  of  copper  20'"'"  long,  IS'^'" 
wide  and  5"^™  thick  weigh  ? 

7.  If  1  lb.  of  rock  salt  is  dissolved  in  1  cu.  ft.  of  water 
without  increasing  its  volume,  what  will  be  the  specific 
gravity  of  the  solution  ? 

8.  How  much  does  a  l)oat  weigh  that  displaces  7000 
cu.  ft.   of  water? 

9.  If  a  boat  is  capable  of  displacing  3000  cu.  ft.,  what 
weight  will  be  required  to  sink  it  ? 


RATIO  AND   PROPORTION  135 

223.    Proportion.     A   proportion  is  an  equality  of  ratios 
and  is  expressed  in  the  following  way : 


a 

e 

a 

:    h: 

=  a  : 

d, 

a  : 

b  : 

:  c  : 

d. 

224.  The  method  of  solving  problems  by  proportion  is 
often  called  the  Rule  of  Three,  since  problems  which  give 
three  quantities  so  related  that  two  of  them  sustain  the 
same  ratio  to  each  other  as  the  third  to  the  quantity 
required,  can  readily  be  solved  by  proportion. 

225.  Thus,  if  any  three  of  the  four  terms  of  a  proportion 
are  known,  the  other  one  can  be  found. 

If  1=^,  then,  :r=3  X  f  =  2f 

01        C  1 ^        c 

Check  by  putting  2\  for  x,  then  =1  =  -,  or  —  =  -. 
J  i  ^7  3       7         21      7 

226.  The  first  and  last  terms  of  a  proportion  are  called 
the  extremes,  and  the  second  and  third  terms  the  means. 

227.  In  any  proportion  the  product  of  the  mean%  is  equal 
to  the  product  of  the  extremes. 

a      c 
If  7  =  -,  then  by  clearing  of  fractions  ad  =  he.     This 
0      d 

•proves  the  proposition,  since  a  and  d  are  the  extremes,  and 
h  and  c  the  means. 

228.  If  1  lb.  of  sugar  costs  4  ct.,  2  lb.  Avill  cost  8  ct. 
and  4  ct.  :  8  ct.  =  1  lb  :  2  lb.  At  the  same  rate  3  lb. 
would  cost  12  ct.,  etc.  The  ratio  of  costs  in  each  case  is 
equal  to  the  ratio  of  the  weights.  The  cost  of  sugar  is 
said  to  be  directly  proportional  to  its  weight. 


136  BATIO  AND  PROPORTION 

Ex.  The  Washington  monument  is  555  ft.  high.  What 
is  the  lieight  of  a  post  that  casts  a  shadow  1  ft.  9  in.  when 
the  monument  casts  a  shadow  192  ft.  6  in.  ? 

Solution  by  proportion. 

Let  X  =  the  height  of  the  post. 

Then  j^^V7d_ 

555      192.5 

111      0.05 

.-.  X  =  W^  X  l.n  =  5.04  ft. 

1.1 

Solution  by  unitary  analysis. 

A  shadow  192  ft.  6  in.  long  is  cast  by  a  monument  555  ft.  high. 

.-.  a  shadow  1  ft.  long  will  be  cast  by  a  post  — — —  ft.  hioh. 

^  ^      ^        192.5         ,  ^-       --- 

.'.  a  shadow  1  ft.  9  in.  long  will  be  cast  by  a  post  -^-^ ^^r—  ft- 

high,  or  5.04  ft.  high.  ^^"-^ 

229.  If  1  man  can  do  a  certain  piece  of  work  in  6  days, 
2  men  working  at  the  same  rate  will  do  the  work  in  3 
days,  3  men  will  do  it  in  2  days,  etc.  2  men  do  the  work 
in  ^  the  time  that  1  man  will  do  it ;  3  men  in  1  the  time, 
etc.  Hence,  as  the  number  of  men  increases,  the  time 
diminishes  in  the  same  ratio.  If  2  men  do  the  work  in  3 
days,  3  men  will  do  it  in  f  of  3  days,  or  2  days.  There- 
fore the  ratio  of  the  number  of  men,  |,  is  equal  to  the 
corresponding  ratio  of  time  inverted.  Hence,  the  number 
of  men  is  said  to  be  inversely  proportional  to  the  time. 

Ex.  The  crew  and  passengers  of  a  steamship  consisted 
of  1500  persons.  The  ship  had  sufficient  provisions  to 
last  12  weeks  when  the  survivors  of  a  wreck  were  taken 
on  board.  The  provisions  were  then  consumed  in  10 
weeks ;   how  many  were  taken  on  board  ? 


RATIO  AND  PROPORTION  137 


Solution  hy  jiroportion. 

Let  X  equal  the  total  number  on  board. 

Then  _^  =  ir, 

1500      10 

or  X  — =  1800, 

and  1800  -  1500  =  300,  the  number  taken  on  board. 

Solution  hy  unitary  analysis. 

There  are  provisions  for  12  weeks  for  1500  persons. 

.-.  there  are  provisions  for  1  week  for  12  x  1500  persons. 

1500  X  1*^ 
.-.  there  are  provisions  for  10  weeks  for  — -^  persons  or  1800 

persons. 

.-.  1800  —  1500  =  300,  the  number  taken  on  board. 


EXERCISE   50 

State  which  of  the  following  are  directly  proportional 
and  which  are  inversely  proportional  : 

1.  The  price  of  bread,  the  price  of  flour. 

2.  The  number  of  workmen,  the  amount  of  work  done 
in  a  given  time. 

3.  The  number  of  workmen,   the  time  required  to  do 
a  given  amount  of  work. 

4.  The  height  of  the  thermometer,  the  temperature. 

5.  The  velocity  of  a  train,  the  time  required  to  go  a 
given  distance. 

6.  The  number  of  horses  bought  for  a  given  sum,  the 
price  per  horse. 

7.  The  price  of  freight,  the  distance  carried. 

8.  The  area  of  a  circle,  the  length  of  its  diameter. 


138  BATIO  AND  PROPORTION 

9.    In  how  many  ways  can  the  terms  of  the  proportion 
2  :  3  =  8  :  12  be  arranged  without  destroying  the  proportion? 

10.  Tlie  assessed  value  of  a  certain  town  is  $7500000, 
and  bonds  for  $6000  are  issued.  What  part  of  tliis  does 
a  person  worth  1 10000  pay? 

11.  A  shadow  cast  by  a  post  6  ft.  higli  is  9  ft.  3  in. 
How  long  is  the  shadow  cast  by  a  church  steeple  150  ft. 
high  ? 

12.  A  merchant  fails  for  $12,300  and  his  property  is 
worth  $5720.  How  much  will  he  pay  a  creditor  Avhom 
he  owes  $2500? 

13.  A  clock  is  set  at  noon  on  Monday;  at  6  p.m.  on 
Wednesday  it  is  2  minutes  and  '  20  seconds  too  slow. 
Supposing  the  loss  of  time  to  be  constant,  what  is  the  cor- 
rect time  when  the  clock  strikes  12  on  Sunday  noon? 

14.  There  are  two  kinds  of  thermometers  used  in  this 
country,  Fahrenheit,  used  to  register  temperature,  and 
Centigrade,  used  largel}^  in  scientific  work.  The  freezing 
point  of  water  is  32°  and  0°  resj^ectively,  while  the  boiling 
point  is  212°  and  100°  respectively.  68°  Fahrenheit  cor- 
responds to  what  temperature  Centigrade  and  5-1°  Centi- 
grade to  what  temperature  Fahrenheit  ? 

15.  There  is  another  kind  of  thermometer  known  as 
Reaumur,  the  freezing  and  boiling  points  being  0°  and  80° 
respectively.  Express  in  Reaumur  scale  70°  on  each  of 
the  other  two. 

16.  The  boiling  point  of  alcohol  is  78°  Centigrade ; 
what  is  the  boiling  point  of  alcohol  on  each  of  the  other 
two  ? 

17.  A  grain  of  gold  can  be  beaten  into  a  leaf  of  56 
sq.  in.  How  many  of  these  leaves  will  make  an  inch  in 
height  if  1  cu.  ft.  of  gold  weighs  1215  lb.? 


RATIO  AND  PROPORTION  139 

18.  Divide  60  into  two  parts  proportional  to  2  and  8. 

19.  Divide  90  into  parts  proportional  to  2,  3  and  4. 

20.  Two  men  start  in  business  with  a  capital  of  fTSOO. 
One  of  them  furnishes  ^4000  and  the  other  18500.  At 
tlie  end  of  a  year  the  profits  are  'i^3250.  How  much  is 
each  man's  share? 

21.  A  man  starts  in  business  with  a  capital  of  85000 
and  in  3  months  admits  a  partner  with  a  capital  of  $>4500. 
At  the  end  of  the  year  the  profits  amount  to  88750. 
How  much  is  each  man's  share  ? 

22.  A  piece  of  work  was  to  have  been  done  by  10  men 
in  20  days,  but  at  the  end  of  two  days  8  men  left.  How 
long  did  it  take  the  remaining  7  men  to  complete  the 
work  ? 

23.  If  the  interest  on  8325  is  872.50  in  a  given  time, 
how  much  is  the  interest  on  8850  for  the  same  time? 

24.  Two  cog  wheels  work  together ;  one  has  36  cogs 
and  the  other  14.  How  many  revolutions  does  the  smaller 
one  make  while  the  larger  one  makes  28  revolutions? 


METHOD   OF   ATTACK 

230.  Ill  solving  any  aritlimetical  problem  tlie  student 
will  find  the  following  suggestions  useful : 

(1)  The  first  essential  is  a  thorough  understanding  of 
the  proper  relations  between  the  conditions  given.  This 
requires  some  form  of  analysis  leading  to  a  complete  state- 
ment of  the  conditions. 

(2)  The  solution  should  involve  no  unnecessary  work. 
Cancellation  and  other  convenient  short  methods  should 
be  used  if  possible. 

(3)  All  arithmetical  work  should  be  carefully  checked. 
The  student  must  realize  tliat  accuracy  is  of  the  highest 
importance  and  that  to  secure  accuracy  his  work  must 
always  be  checked.  Any  arithmetical  work  that  has  an 
error  in  it  is  valueless.  The  check  also  gives  the  student 
a  means  of  knowing  for  himself  whether  he  has  a  correct 
result  or  not.     He  has  no  need  of  answers  to  his  problems. 

Ex.  1.  If  the  time  of  the  beat  of  a  pendulum  varies 
as  the  square  root  of  its  length,  and  the  length  of  a  pendu- 
lum that  beats  seconds  is  39.2  in.,  find  the  length  of  a 
pendulum  that  beats  50  times  a  minute. 

Solution.  The  given  pendulum  beats  GO  times  per  minute,  the 
required  pendulum  beats  50  times  per  minute. 

Since  the  longer  the  pendulum  the  more  slowly  it  beats,  the  re- 
quired pendulum  is  longer  than  the  given  one. 

Therefore,  the  square  root  of  the  lengths  of  the  pendulums  are 
in  the  ratio  f§,  or  ^. 

140 


METHOD   OF  ATTACK  141 

Let  I  =  the  length  of  the  required  pendidum. 

-I  hen,  — ; =  -, 

V;39.2      5 

I        62 

or  =  — , 

39.2      52' 

;      6  X  6  X  39.2  .  6  X  6  X  39.2  x  4  .  -«  1,0  • 

or  I  = in.  = in.  =  00.448  in. 

5x5  100 

Check  either  by  changing  the  order  of  the  factors  and  performing 
the  multiplication  again,  or  by  casting  out  the  nines. 

Ux.  2.  The  greatest  possible  sphere  is  cut  from  a  cube, 
one  of  whose  edges  is  3  ft.  Find  the  portion  of  the  cube 
cut  away. 

Solution.     The  volume  of  the  cube  is  3^  cu.  ft. 
The  volume  of  the  sphere  is  f  tt  x  (|)^  cu.  ft. 
Therefore  the  portion  cut  away  is  3^  cu.  f t.  —  |  tt  x  (f  )^  cu.  ft. 
Without  performing  the  operations  indicated  the  student  can  by 
cancellation  and  combination  of  terms  v^rite  the  result  thus, 

3-2^3  _?^cu.ft.  =  32(?^I^^^^cu.ft.  =  9x  1.4292  cu.ft.  =  12.8628  cu.  ft. 

Check  as  before. 

Ex.  3.  Find  the  area  of  a  square  field  whose  diagonal 
is  50  rods. 

Solution.     Let  x  =  one  side  of  the  square  field. 

Then  x^  +  x^  =  bO% 

or  2  x2  =  502. 

502 
.-.  x^,  or  the  area  of  the  field  in  square  rods,  =  '-—  sq.  rd.  =  7|f  acres. 

Check  each  step  in  the  work. 

Ux.  4.  Find  the  area  of  the  circle  which  is  equal  in 
area  to  two  circles  whose  radii  are  5  in.  and  7  in. 


142  METHOD   OF  ATTACK 

Solution.     Let  r  =  the  radius  of  the  required  circle. 

Then  its  area  in  square  inches  =  irr'^  =  tt  x  5-  +  tt  x  7^  =  7r(52  +  7^) 
=  TT  X  74,  or  232.48  sq.  in. 

Check  each  step  in  the  work. 

Here,  instead  of  multiplying  tt  by  25  and  then  by  49  and  adding 
the  results,  time  is  saved  by  adding  25  and  49  and  multiplying  tt  by 
the  sum,  74. 

231.  The  foot  pound  is  used  as  a  unit  of  work.  This 
unit  is  defined  as  tlie  amount  of  work  required  to  over- 
come the  resistance  of  one  pound  through  a  space  of  one 
foot.  The  rate  of  work  is  generally  defined  by  using  the 
term  horse  poiver.  An  engine  of  one  horse  power  can  do 
33000  foot  pounds  of  work  in  one  minute,  i.e.  can  over- 
come a  resistance  of  33000  pounds  through  a  space  of  one 
foot  in  one  minute. 

Ex.''5.  AVhat  horse  power  is  an  engine  exerting  that 
draws  a  train  with  a  uniform  speed  of  40  miles  an  hour 
against  a  resistance  of  1000  pounds  ? 

Soluiion.  The  amount  of  work  done  in  one  hour  is  1000  x  40 
X  5280  foot  pounds. 

rrx.  ^     f         1    J         •  •      ^    '    1000  X  40  X  5280  .     ^ 

The  amount  of  work  done  in  one  minute  is foot 

1  60 

pounds.  2       ig 

rr.        i         ,,          -PI-              1    •    1000  X  ^0  X  ^2^0  , 
Therefore,  the  rate  of  doing  work  is — -  horse  power 

^p  X  nm 

3 

= x_j-2< horse  power  =  106|  horse  power. 

Check  each  step. 

232.  The  student  will  notice  that  in  each  of  the  above 
exercises,  first,  the  relations  between  the  given  conditions  are 
carefully  established ;  and  second,  a  complete  statement  of 
these  conditions  is  frriften  out  and  the  work  shortened  as  mucli 
as  possible  by  cancellation  or  otherwise^  before  the  processes 


METHOD   OF  ATTACK  143 

of  multiplication  and  dividon  are  used.  Frequently  stu- 
dents in  solving  such  problems  will  perform  tlie  operations 
indicated  at  each  step,  thus  doing  a  hxrge  amount  of  un- 
necessary work.  By  carefully  studying  these  model  solu- 
tions the  student  will  see  where  the  unnecessary  work  can 
be  avoided. 

As  indicated  in  Art.  41,  it  is  a  good  plan,  whenever  pos- 
sible, to  estimate  the  result  mentally  and  to  compare  this 
rough  estimate  with  the  result  found  by  solving  the  prob- 
lem. This  will  prevent  large  errors  and  such  errors  as 
arise  from  misplacing  the  decimal  point. 

EXERCISE   51 

1.  Find  the  area  bounded  by  6  eqaal  coins  whose 
centers  are  at  the  vertices  of  a  regular  hexagon,  the  diam- 
eter of  each  coin  being  2.38'^'". 

2.  A  crescent  is  bounded  by  a  semi-circumference  of  a 
circle  whose  radius  is  15  inches,  and  by  the  arc  of  another 
circumference  whose  center  is  on  the  first  arc  produced. 
Find  the  area  and  perimeter  of  the  crescent. 

3.  A  horse  is  tied  with  a  50  ft.  rope  to  one  corner 
of  a  barn  30  ft.  by  40  ft.  Find  the  area  he  can  graze 
over. 

•  4.  A  well  30  ft.  deep  and  4  ft.  in  diameter  is  to  be 
dug.  If  a  cubic  foot  of  earth  weighs  12  lb.,  how  much 
work  is  to  be  done? 

5.  A  horse  drawing  a  wagon  along  a  level  road  at  tlie 
rate  of  2  mi.  an  hour  does  29216  foot  pounds  of  work 
in  3  min.  What  pull  in  pounds  does  he  exert  in  drawing 
the  wagon? 


144  METHOD   OF  ATTACK 

6.  A  uniform  heavy  bar,  12  ft.  long  and  weighing  80 
lb.,  rests  on  2  props  in  the  same  horizontal  plane,  so  that 
2  ft.  projects  over  one  of  the  props ;  find  the  distance  be- 
tween the  props  so  that  the  pressure  on  one  may  be  double 
that  on  the  other ;  also  find  the  pressures. 

7.  It  is  proved  in  geometry  that  similar  volumes  are 
to  each  other  as  the  cubes  of  their  like  dimensions.  If 
a  cubical  bin  whose  edge  is  4  ft.  holds  52  bu.  of  wheat, 
how  many  bushels  will  a  bin  6  ft.  on  an  edge  hold? 

8.  The  temperature  remaining  the  same,  the  space  oc- 
cupied by  a  gas  varies  inversely  as  the  pressure.  At  a  con- 
stant temperature  a  mass  of  air  occupies  25  cu.  ft.  under 
a  pressure  of  10  lb.  to  the  square  inch ;  what  space  will 
it  occupy  under  a  pressure  of  26  lb.  to  the  square  inch? 

9.  A  cubic  foot  of  water  weighs  1000  oz.,  and  the 
pressure  of  the  air  is  336  oz.  per  square  inch ;  find  the 
pressure  on  a  square  foot  at  a  depth  of  10  ft.  below 
the  surface  of  a  pond. 

10.  If  the  specific  gravity  of  mercury  is  13.598  and 
the  weight  of  a  cubic  inch  of  water  is  252.6  grains,  find 
the  pressure  of  air  per  square  inch  in  pounds  when  the 
mercury  in  the  barometer  stands  at  30.5  in. 

11.  An  iceberg  (specific  gravity  0.925)  floats  in  sea 
water  (specific  gravity  1.025).  Find  the  ratio  of  the  part 
out  of  water  to  the  part  immersed. 

12.  A  piece  of  lead  placed  in  a  cylindrical  vessel,  the 
radius  of  whose  base  is  1.2^"^,  causes  the  liquid  in  the 
vessel  to  rise  3^"™.  What  is  the  volume  of  the  piece  of 
lead,  and  how  much  does  it  weigh  if  lead  is  11.2  times  as 
heavy  as  water  ? 


MISCELLANEOUS  EXERCISE  145 

MISCELLANEOUS   EXERCISE   52 

Express  the  ratio  of  : 

1.  A  cubic  decimeter  to  a  liter. 

2.  A  cubic  centimeter  to  a  cubic  uiillimeter. 

3.  A  cubic  decimeter  to  a  cubic  meter. 

4.  A  kilogram  to  a  centigram. 

5.  A  meter  to  a  yard. 

6.  A  quart  to  a  liter. 

7.  A  kilogram  to  a  pound. 

8.  A  milligram  to  a  kilogram. 

9.  A  kilogram  to  40  grams. 

10.    A  kilometer  to  200  centimeters. 

Find  the  value  of : 


11. 

(60--i/)x3. 

13. 

(W-5)x6. 

12. 

120         ^ 
12  x  50 

14. 

(3|9  +  2)  x4. 

15. 

/522      2727      144 

8x9^, 

12 

V  6          22        180 

4x57'^ 

16. 

(26-13)x7 
2+15-3 

18. 

17. 

5  X  8-17  X  2 
17-14 

19. 

Ti:+3i    ^j       31 
If              lixf 

20.  What  is  a  decimal  fraction  ? 

21.  How  is  the  units'  place  distinguished  ? 

22.  What  is  the  place  value  of  a  digit  one  place  to  the 
right  of  units  ?  three  places  to  the  right  ? 


lyman's  adv.  ar. — 10 


146  MISCELLANEOUS  EXERCISE 

23.  What  is  the  importance  of  the  symbol  0  in  the  deci- 
mal scale  of  notation  ? 

24.  If  a  decimal  fraction  is  multiplied  by  a  digit  in  units' 
place,  do  the  place  values  of  the  digits  in  the  product  dif- 
fer from  the  place  value  of  the  digits  in  the  multiplicand  ? 
If  the  decimal  fraction  is  multiplied  by  the  same  digit  two 
orders  lower,  is  there  a  difference  in  the  place  value  of  the 
digits  in  the  product  ? 

25.  If  a  decimal  fraction  is  divided  by  a  digit  in  units' 
place,  do  the  place  values  of  the  digits  in  the  quotient  dif- 
fer from  the  place  values  of  the  digits  in  the  dividend? 
If  the  decimal  fraction  is  divided  by  the  same  digit  three 
orders  higher,  what  is  the  difference  in  the  place  values  of 
the  digits  in  the  quotient  ? 

26.  What  is  a  divisor  of  a  number  ?  a  common  divisor 
of  two  or  more  numbers  ?  the  greatest  common  divisor  of 
two  or  more  numbers  ? 

27.  What  is  a  multiple  of  a  number  ?  a  common  multi- 
ple of  two  or  more  numbers  ?  the  greatest  common 
multiple  of  two  or  more  numbers  ? 

28.  What  is  a  prime  number  ?  What  is  a  prime  factor  of 
a  number  ?     When  are  two  numbers  prime  to  each  other  ? 

29.  What  is  the  shortest  piece  of  rope  that  can  be  cut 
exactly  into  pieces  12,  15  and  20  ft.  long  ? 

30.  Find  the  1.  c.  m.  of  the  first  five  odd  numbers,  also 
of  the  first  six  even  numbers. 

31.  Find  the  g.  c.  d.  of  125,  340  and  735. 

32.  Evaluate  3|  +  5^  4-  7^  +  9^. 

33.  Evaluate  |  +  ^^  +  iV  +  2V  "  I  -  I'o  "  A- 

34.  Evaluate  4^^  +  2  x  5f  -  3  x  |  -  i. 


MISCELLANEOUS   EXERCISE  147 

35.  A  cubic  foot  of  water  weighs  1000  oz.  How  many 
tons,  etc.,  of  water  are  there  in  a  canal  80  ft.  wide,  8  ft. 
deep  and  10  mi.  long  ? 

36.  How  niany  feet  per  second  are  equal  to  40  mi.  an 
hour  ? 

37.  Find  the  square  root  of  0.4 ;   the  cube  root  of  0.27. 

38.  If  I  walk  7.2^'"  in  1  lir.,  how  far  shall  I  go  in  6  hr. 
and  20  min.  at  the  same  rate  ? 

39.  How  many  cubic  centimeters  of  air  are  there  in  a 
room  9^'"  long,  6  J'"  wide  and  3.15™  high  ? 

40.  What  is  the  area  of  a  cube  that  has  the  same  volume 
as  a  box  2  ft.  6  in.  by  2  ft.  3  in.  by  2  ft.  ? 

41.  How  many  cubic  meters  of  water  pass  under  a  bridge 
in  one  minute  when  the  river  is  20™  wide,  4™  deep  and  is 
running  3^™  per  hour? 

42.  Write  three  numbers  of  four  figures  each  that  are 
divisible  by  both  8  and  3. 

43.  Write  three  numbers  of  six  figures  each  that  are 
divisible  by  both  9  and  11. 

44.  Replace  the  zeros  in  205006  so  that  the  number  may 
be  divisible  by  both  9  and  11. 

45.  What  is  the  cost  per  hour  of  lighting  a  room  with 
40  burners,  each  consuming  2 J  cu.  in.  of  gas  per  second, 
the  price  of  gas  being  ^1.25  per  thousand  cubic  feet? 

46.  A  roller  used  in  rolling  a  lawn  is  6|  ft.  in  circum- 
ference and  2|^  ft.  wide.  If  the  roller  makes  10  revolutions 
in  crossing  the  lawn  once  and  must  pass  back  and  forth  12 
times  to  cover  the  whole  lawn,  find  the  area  of  the  lawn. 

47.  Find  the  sum  of  J  +  J  +  2%  +  3%  correct  to  four  deci- 
mal places. 


148  MISCELLANEOUS  EXERCISE 

48.  Find  each  of  the  following  products  correct  to  five 
significant  figures : 

(a)  20.361  X  40.482.      (h)  1.5674  x  75.429. 

(0  824.763  X  45.  {d)  103.64  x  0.033. 

(e)  0.423x0.00765. 

49.  YmA  each  of  the  following  quotients  correct  to  0.01 : 

(a)  22-3.1416;   (h)  42.567  h- 21.268 ;    «)  0.4-0.75; 
(cZ)  237.64-2.1473.;    (e)  2-9.97. 

50.  Find  the  cost  of  carpeting  a  room  12  ft.  3  in.  long  and 
10  ft.  9  in.  wide  with  carpet  27  in.  wide  at  $1.12  a  yard. 

51.  Find  the  cost  of  8  T.  1450  lb.  of  coal  at  17.25  a  ton. 

52.  Multiply  7644  by  331  and  divide  the  result  by  16f. 

53.  Divide  8350  by  25  and  multiply  the  result  by  12i. 

Find  the  value  of : 

54.  0.0001x0.0001;  6.74x21.023. 

55.  1.1  X  0.011;  7.6  xO.76. 

56.  2.5  X  25  X  250 ,  2.5  x  0.25  x  0.025. 

57.  0.002  X  3.01 ;  0.0005  x  0.01  x  5000000. 

58.  15.625-25;  0.15625^2.5. 

59.  8-0.002;   50-0.25. 

60.  9.065-0.049;  0.005-0.01. 

61.  0.00128-8.192;  1708.4592-0.00024. 

Find  correct  to  4  decimal  places  : 

62.  0.138138  +  0.1425876  +  2.060606  +  0.008964. 

63.  7.427525-2.347596.         65.    0.33i-0.37|. 

64.  0.331  X0.37J.  66.   0.0404^7692. 


MISCELLANEOUS  EXERCISE  149 

67.  If  the  length  of  Jupiter's  day  is  9  hr.  56  min.,  how 
many  more  days  has  Jupiter  than  the  earth  in  one  year? 

68.  If  i500  can  be  counted  in  one  minute,  how  long  will 
it  take  to  count  -$1000000  ? 

69.  What  is  the  difference  between  the  daily  income 
of  a  man  whose  salary  is  $1200  a  year  and  of  one  wJiose 
salary  is  $1600? 

70.  Counting  12  hr.  a  day,  liow  long  would  it  take  to 
count  a  billion  at  the  rate  of  750  a  minute  ? 

71.  How  many  days  old  was  a  person  Oct.  5,  1904,  who 
was  born  July  27,  1861  ? 

72.  The  ancient  Roman  mile  is  0.917  of  the  English 
mile.  Express  the  diameter  of  the  earth  (7926  English 
miles)  in  Roman  miles. 

73.  The  diameter  of  a  fly  wheel  is  found  by  measure- 
ment to  be  20.12  in.     Find  its  circumference. 

74.  The  specific  gravity  of  copper  is  8.97;  of  gold, 
19.36  ;  of  lead,  11.36.  Find  the  weight  of  a  lump  of  each 
equal  in  bulk  to  a  liter  of  water. 

75.  The  diameter  of  the  earth  is  7926  mi.  The  sun's 
diameter  is  111.454  times  the  earth's  diameter.  Find  the 
sun's  diameter  correct  to  miles. 

76.  A  lump  of  iron  containing  12  cu.  ft.  is  drawn  out 
into  a  rod  50  ft.  long.     What  is  the  diameter  of  the  rod  ? 

77.  The  true  length  of  the  year  is  365.2426  da.  What 
error  is  made  by  calculating  the  year  as  365  da.,  and  add- 
ing a  day  every  leap  year,  omitting  three  leap  years  in 
four  centuries  ? 

78.  The  edge  of  a  cube  is  12  in.  What  is  the  edge  of  a 
cube  three  times  its  volume  ? 


150  MISCELLANEOUS  EXERCISE 

79.  How  many  miles  an  hour  does  a  person  walk  who 
takes  two  steps  a  second  and  1900  steps  to  the  mile  ? 

80.  Express  in  words  0.12071  and  12000.00071. 

81.  How  many  steps  0.8  of  a  meter  long  Avill  a  person 
take  in  walking  10^"*  ? 

82.  A  clock  which  gains  one  minute  in  10  hr.  is  correct 
on  Monday  noon.  What  is  the  correct  time  when  it  indi- 
cates Monday  noon  of  the  next  week  ? 

In  scientific  work,  when  numbers  depend  upon  measurements  and 
therefore  cannot  be  expressed  with  absolute  accuracy  the  index  nota- 
tion is  frequently  used.  Thus,  the  wave  length  of  blue  light,  deter- 
mined by  the  physicist  to  be  0.000431'"'"  would  usually  be  written 
4.31  X  lO""*'"*^.  The  distance  from  the  sun  to  the  earth  is  determined 
by  the  astronomer  to  be  approximately  93000000  mi.  In  index  nota- 
tion it  would  be  written  9.3  x  10^  mi. 

83.  Express  the  following  in  the  index  notation  : 
0.0000025;    36500000000;    2000V000;    41100000. 

84.  Express  in  the  common  notation  1.1  x  10~^;  3.6  x  10^; 
4.321x10-8;  5x10-4;  5x106. 

85.  From  3542g  subtract  2131g. 

86.  Find  the  sum  of  34. 6^2,  121. 51^2,  and  25.11i2  and 
express  it  in  the  decimal  notation. 

87.  If  brass  weighs  525  lb.  per  cubic  foot,  find  the 
weight  of  a  circular  brass  plate  21  in.  in  diameter  and 
^  in.  thick. 

88.  If  a  cubic  foot  of  gold  may  be  made  to  cover  uni- 
formly 432000000  sq.  in.,  tind  the  thickness  of  the  gold. 

89.  If  a  gallon  of  water  contains  277.274  cu.  in.,  and  a 
cubic  foot  of  water  weighs  1000  oz.,  liow  much  does  a  pint 
of  water  weigh  ?     How  many  gallons  will  weigh  a  ton  ? 


MISCELLANEOUS  EXERCISE  151 

90.  Four  circles  each  1  ft.  in  diameter  are  so  placed  tliat 
two  of  them  touch  two  of  the  others,  and  the  remaininof 
two  both  touch  three  of  the  others;  find  the  area  of  the 
figure  whose  angles  are  at  the  four  centers. 

91.  What  (standard)  time  is  it  in  Boston  when  it  is 
4.30  P.M.  in  San  Francisco? 

92.  A  ship's  clock  is  corrected  at  1  o'clock  each  day. 
If  the  ship  passes  over  10°  30'  each  day,  wliat  change  must 
be  made  in  the  clock  (a)  if  the  ship  is  sailing  from  W. 
to  E.  ;   (/))  from  E.  to  W.  ? 

93.  Find  the  remainders  (without  dividing)  after  471321 
has  been  divided  by  all  of  the  numbers  (except  7)  from  2 
to  12  inclusive. 

94.  Show  without  dividing  that  133056  is  divisible  by 
792. 

95.  A  ship's  clock  is  corrected  every  day  at  1  p.m.;  how 
much  must  it  be  put  back  or  forward  at  1,  if  the  ship  has 
passed  over  11°  of  longitude  from  east  to  west  ? 

96.  When  it  is  noon  (standard  time)  Wednesday,  Dec.  7, 
at  Chicago,  what  time  and  date  is  it  at  Rome  ?  at  Tokyo  ? 

97.  A  meter  is  defined  as  1  x  lO"*"  of  the  distance  from 
the  pole  to  the  equator.  Find  the  circumference  of  the 
earth  in  kilometers. 

*    98.    Find  the  circumference  of  the  earth  in  miles  if  the 
meter  is  equal  to  39.37079  in. 

99.  If  1  cu.  ft.  of  water  weighs  1000  oz.,  and  platinum 
is  20.337  times  as  heavy  as  water,  how  many  feet  of 
platinum  wire  g^-o^Q-Q  of  an  inch  in  diameter  would  weigh 
a  grain  ? 


PEKCENTAGE 


233.       I  =  j%  =  0.50  =  50%  =  50  per  cent, 


33 


and  J  =  ^  =  0.331  =  331  %  =  331  per  cent. 

These  are  different  ways  of  denoting  the  same  fractional 
part.  In  business  operations  it  is  customary  to  express 
fractions  in  hundredths,  but  in  stating  problems  the 
denominator  100  is  omitted  and  the  per  cent  symbol,  %, 
or  the  expression  per  cent  is  used.  Percentage  is  there- 
fore only  an  apj)lication  of  the  decimal  fraction  and  not 
a  separate  department  of  arithmetic. 

234.  The  word  percentage  is  derived  from  the  Latin 
2)er  centum,  meaning  bi/  the  htmdredths. 

235.  The  number  denoting  how  many  hundredths  are 
taken  is  called  the  rate  per  cent.  Thus,  if  5%  of  a  number 
is  to  be  taken,  5  is  called  the  rate  per  cent,  and  5%  the  rate. 

236.  The  following  examples  illustrate  several  closely 
related  operations  frequently  used  in  business  transactions. 

Ux.  1.   What  is  8%  of  -f750? 
Solution.     8  %  of  $750  =  0.08  of  |750  =  |60. 

Ux.  2.    12  is  what  per  cent  of  240  ? 
Solution.     Let  x  %  =  the  rate. 

Then  a:  %  of  240  =  12, 


152 


PERCENTAGE  158 


Ex.  3.    20  is  G%  of  wliiit  nun-iber  ? 
Solution.     Let  x  =  the  n umber. 

Then  6  %  of  a;  =  20, 


0.06  ^ 


EXERCISE   53 

1.  Express  the  following  fractions   in  per   cent,   also 

n«   rlppimnU-     13        9       51211312. 
db   aeoillldib  .    "2",    p     10'    6'    "8'     5^'    T'    10^'    ^~' 

2.  3  is  what  per  cent  of  4  ?  8  is  what  per  cent  of  4  ? 
18  is  what  per  cent  of  27  ?  25  is  what  per  cent  of  200  ? 
7  is  what  per  cent  of  2  ? 

3.  The  population  of  a  town  is  7250.  What  is  the 
population  live  years  later  if  it  has  increased  7%  in  that 
time  ? 

4.  A  town  of  11750  inhabitants  decreases  12%  in  ten 
years.     What  is  its  population  after  this  loss  ? 

5.  Express  the  following  as  decimals:  |%,  331%, 
0.5%,  125%. 

6.  What  is  -1%  of  75?  1%  of  100?  0.1%  of  -|  ? 
f%ofi^? 

7.  Write  as  per  cent  l-J,  2|,  i  O.OOl  10,  2,  0.25,  2.5, 
0.16|. 

8.  Tlie  attendance  in  a  certain  school  increased  in  one 
year  from  318  to  425 ;  find  the  rate  per  cent  of  increase. 

9.  In  a  certain  school  there  are  291  boys  and  315  girls. 
What  percentage  of  the  attendance  is  boys  and  what  per- 
centage is  girls  ? 

10.  In  a  certain  town  the  total  school  enrollment  is  962  ; 
of  this  156  are  in  the  high  school.  What  percentage  of 
the  whole  enrollment  is  in  the  high  school  ? 


154  PERCENTAGE 

11.  If  0.8%  of  those  living  at  the  age  of  24  die  within 
a  year,  how  many  out  of  6625  persons  of  this  age  die 
during  that  period  ? 

12.  At  the  age  of  15,  735  out  of  96285  die  within  a 
year.     Wliat  is  the  rate  per  cent  of  deaths  ? 

13.  At  the  age  of  25,  718  out  of  89032  die  within  one 
year.  Is  the  death  rate  higher  or  lower  than  at  the  age 
of  15? 

14.  A  man  owns  a  farm  worth  -^7500.  His  annual  taxes 
are  -$68.50.  How  much  must  he  make  in  order  to  clear 
6  %  from  his  farm  each  year  ? 

15.  A  house  depreciates  in  value  each  year  at  the  rate 
of  5%  of  its  value  at  the  beginning  of  the  year,  and  its 
value  at  the  end  of  three  years  is  §4225;  find  the  original 
value. 

16.  A  man  sold  two  horses  for  f  200  each;  on  the  pur- 
chase price  of  one  he  made  20%,  and  on  the  other  he  lost 
25%.     Did  he  gain  or  lose  and  how  much  ? 

17.  The  wholesale  grocer  buys  coffee  at  25  ct.  per 
pound  and  sells  it  at  30  ct.  The  retail  grocer  bu^-s  it 
at  30  ct.  and  sells  it  at  Sl^  ct.  What  per  cent  does  each 
make  ? 

18.  If  a  person  spends  60%  of  his  income  and  saves 
i  1000,  what  is  his  income  ? 

19.  Which  investment  returns  the  larger  per  cent,  flour 
costing  i  1.98  per  hundred  pounds  and  sold  for  $2.10,  or 
sugar  costing  3J  ct.  a  pound  and  sold  for  4-|-  ct.  ? 

20.  A  man  owning  a  f  interest  in  a  store  sold  ^  of  his 
interest.  What  per  cent  of  his  share  did  he  sell,  and  what 
per  cent  of  the  store  did  he  still  own  ? 


PJtRCENTAGE 


155 


21.  A  mercliant  sold  out  his  stock  of  goods  at  a  discount 
of  10%  of  the  cost  and  realized  't>14T56.84.  How  much 
did  his  goods  cost  him  ? 

22.  A  house  rents  for  #300  a  year,  which  represents  6% 
of  its  value.     How  much  is  it  worth  ? 

23.  In  1880  the  population  of  the  United  States  was 
50152866,  in  1890  it  was  63069756,  and  in  1900  it  was 
75994575.  During  which  decade  was  the  per  cent  of  in- 
crease greater  and  hoAV  much  ? 

24.  What  is  the  difference,  in  square  yards,  between  | 
of  an  acre  and  |  %  of  an  acre  ? 

25.  The  population  of  a  city  is  14560,  and  is  35%  more 
tlmn    it    was    10    yr.    ago. 
then  ? 


What    was    the    population 


26.  On  Nov.  1, 1897,  the  amount  of  money  in  circulation 
in  the  United  States  was :  gold  (including  gold  certifi- 
cates), -1^576000000;  silver  (including  silver  certificates), 
1496000000;  paper,  .15634000000.  Nov.  1,  1902,  the  cor- 
responding amounts  were  1967000000,  $623000000  and 
1736000000.  What  was  the  per  cent  of  increase  in  each 
case  during  the  5  ji\^  and  what  was  the  total  per  cent  of 
increase  ? 

27.  The  following  tables  show  the  total  receipts  and 
disbursements  of  three  of  the  largest  life  insurance  com- 
panies in  the  United  States  for  the  year  1902 : 


Total  Income 

Expenses  and 
Taxes 

Death  Claims 

Other  Disburse- 
ments 

1073636984 
782424835 
330651136 

183485217 

156329328 

54403289 

252617938 

163663466 

60459793 

316541543 

185702274 

69056722 

156  PERCENTAGE 

Find  the  per  cent  of  the  total  income  remaining  in  the 
hands  of  each  compan}^  at  the  end  of  the  year.  Find  the 
per  cent  of  expense  to  income  and  of  death  claims  to 
income  in  each  case. 

28.  In  1890  the  total  foreign  population  in  the  United 
States  was  9249547,  of  whom  2784894  were  born  in  Ger- 
many and  1871509  in  Ireland.  The  population  of  the 
United  States  in  1890  being  63069756,  what  per  cent  of 
the  population  was  born  in  Germany,  and  what  per  cent 
in  Ireland  ? 

29.  In  1890  the  total  number  of  negroes  in  the  United 
States  was  7470000,  which  was  11.8%  of  the  total  popula- 
tion at  that  time.  Determine  the  population  correct  to 
thousands. 

30.  In  1898  the  total  value  of  the  exports  from  the 
United  States  was  sfi^  1231482330,  the  total  value  of  im- 
ports was  '1616049654.  By  what  per  cent  did  the  value 
of  the  exports  exceed  the  value  of  the  imports  ? 


COMMERCIAL   DISCOUNTS 

237.  Manufacturers,  publishers  and  wholesale  dealers 
have  a  fixed  price  list  for  their  products.  Their  customers 
are  allowed  certain  discounts  from  their  list  price,  deter- 
mined by  the  current  market  value.  Thus,  a  book  may 
be  published  at  $1.50  with  a  discount  of  20^  to  dealers. 
The  $1.50  is  the  list  price  and  20  fo  is  the  discount.  Tlie 
list  price  less  the  discount  ($1.50  —  20^  of  $1.50  =  $1.20) 
is  the  net  price,  or  cost. 

238.  To  avoid  the  inconvenience  and  expense  of  issuing  a  new- 
catalogue  whenever  the  market  values  change,  business  houses  gen- 
erally print  a  new  trade  price  list  giving  new  discounts,  without 
issuing  a  new  catalogue.  The  discount  is  changed  either  by  increas- 
ing or  diminishing  the  single  rate  of  discount  already  allowed,  accord- 
ing as  the  cost  of  production  is  diminished  or  increased.  If  the 
discount  is  to  be  increased,  the  change  is  generally  made  by  quoting 
a  further  discount.  Thus,  in  a  catalogue  of  electrical  goods  a  32 
candle  power  lamp  is  quoted  at  ^1.20.  In  trade  price  list  j\.,  accom- 
panying the  catalogue,  a  discount  of  .50  %  is  allowed  on  small  orders. 
In  trade  price  list  B,  issued  later  on  account  of  a  change  in  the  cost 
of  production,  a  discount  of  50%  and  15%  is  allowed.  A  dealer  buy- 
ing 10  lamps  according  to  trade  price  list  A  would  pay  10  x  f  1.20 
-  50%  of  10  X  ^1.20  =  16,  while  according  to  trade  price  list  B  he 
would  pay  $6  -  15%  oi  $Q  =  $5.10. 

The  discount  is  frequently  increased  in  case  of  large  orders.  Thus, 
in  the  above  trade  price  list,  a  discount  of  50  %  is  allowed  on  all  orders 
for  less  than  25  lamjxs,  a  discount  of  50  %  and  20  %  is  allowed  on  all 
orders  for  25  to  100  lamps,  and  a  discount  of  50%,  20%  and  10%  on 
orders  for  100  or  over. 

157 


158 


COMMER  CIA  L  DISCO  UN  IS 


239.  Bills  are  generally  made  out  payable  in  30,  60  or 
90  days,  subject  to  a  certain  discount  for  cash,  or  if  paid 
before  due.  Business  houses  usually  print  on  their  bill 
heads  their  terms  of  discount  for  cash,  e.g.  "  Terms  :  60 
days,  or  2%  discount  for  cash."  "Terms:  net  90  days, 
or  3%   in  10  days." 

Ux.  1.  On  March  12,  1903,  E.  C.  Horner  &  Co.  bought 
of  James  Bros.,  Chicago,  50  plows,  listed  at  §6.50,  less 
25%  and  10%.     Terms:   90  days,  3%  in  10  days. 


E.  C.  HoRXER  &  Co. 


Terms:  OOdaj^s;  3%  10  days. 


Bill  Rendered 

Chicago,  III.,  March  12,  1903. 

Bought  of  James  Bros. 


50  Plows 

@  $6.50 
Discount,  25% 

Discount,  10% 

$325 
81 

00 

25 

243 
24 

75 
38 

219 


37 


If  Horner  &  Co.  avail  themselves  of  cash  payment,  they  will  deduct 
3%  of  $219..37  =  ^6.58,  and  send  the  remainder,  $212.79,  to  James 
Bros.  If  the  bill  is  not  j)aid  till  the  90  days  expire,  they  will  send 
$219.37. 

Ux.  2.    Find  the  cost  of  a  bill  of  goods  amounting  to 
i75  less  20%,  5%  and  2%  for  cash. 
Solution.     Let     x  =  the  cost. 

Then  x  =  0.98  x  0.95  x  0.80  of  $  75  =  $  55.86. 

Atrnli/fils.     $75  is  the  list  price. 

Then  $75  -  20%  of  $75  =  0.80  of  $75  is  the  amount  left  after  the 
first  discount.  And  0.80  of  $75  -  5%  of  0.80  of  $75  =  0.95  x  0.80  of 
$75  is  the  amount  left  after  the  second  discount.     And  0.95  x  0.80  of 


COMMERCIAL  DISCOUNTS  159 

175-2%  of   0.95x0.80   of   .| 75  =  0.98  x  0.95  x  0.80   of   i$75  is   the 
amount  left  after  the  third  discount. 

.-.  0.98  X  0.95  X  0.80  of  |75  =  |55.8G  is  the  net  price  or  cost. 

Second  Sohition.     5)$  75  =  list  price. 

1 15  =  20%  discount. 

2())p0 

^■]  =  5%  discount 
50)1 57 

^    1.14  =  2%  discount  for  cash. 

$  55.86  =  cost  of  the  goods. 

I^x.  3.  What  must  be  the  list  price  of  goods  in  order  to 
reahze  $243  after  deducting  discounts  of  25j/o,  10  fo  and 
10  fo'? 

Solution.     Let  x  =  the  list  price. 

Then  0.90  x  0.90  x  0.75  of  a:  =  |243. 

3         400 
.    ^  _  !|  243         _  ^  m  X  190^0  _  1 400. 


0.90  x  0.90  X  0.75  ^19  x  /7^ 

EXERCISE   54 

1.  Find  the  net  amount  of  the  bill  to  render  in  each  of 
the  following  cases : 

(a)  .|T50  less  33i/o. 
lb)  -$1250  less  25/g  and  15fo. 
(c)  1525  less  20^6,  10  fo  and  5fo. 
Id)  1525  less  3%  10  fo  and  20  fo. 
(e)  $5050.75  less  oO  fo  and  10  fo. 

2.  March  1,  1903,  tlie  iManhattan  Electrical  Supply  Co. 
sold  George  J.  Fiske  &  Co.  the  following  bill  of  goods, 
60  da.,  2fo  10  da.  :  2  electrical  gongs  at  $17.22  each,  less 
40^0  and- lO fo;  2  hotel  annunciators  at  $  15  each,  less  60 fo : 
2  spools  of  wire  at  75  ct.  each,  less  50  fo  and  10^.  Find 
the  amount  to  be  remitted  if  paid  March  11,  and  write  the 
bill  rendered. 


160  COMMERCIAL  DISCOUNTS 

3.  A  piano  listed  at  $  750  was  sold  at  a  discount  of 
40  fo  and  10  fo.  If  the  freight  was  84.87  and  drayage  $3, 
what  was  the  net  cost  of  the  piano  ? 

4.  Find  the  net  cost  of  a  piece  of  Rogers's  statuary 
listed  at  $65  and  discounted  at  35^,  20  fo,  10  fc  and  5fo. 

5.  A  merchant  buys  f  1750  worth  of  goods  at  a  discount 
of  331/0  and  10  fo.  If  he  sells  the  goods  at  the  list  prices, 
what  is  the  rate  of  gain  on  the  cost  ? 

6.  A  car  load  of  flour  weighing  195  hundredweight 
cost  a  grocer  $  1.85  a  hundredweight.  If  he  is  allowed  a 
discount  of  1^  for  cash  and  sells  the  flour  for  $2.10  a 
hundredweight,  how  much  does  he  make  ? 

7.  Which  is  the  greater,  a  discount  of  10^,  10%  and 
10/0,  or  a  discount  of  20  fo,  bfo  and  5fo? 

8.  A  merchant  buys  goods  at  a  discount  of  -10  fo  and 
10  fo  and  sells  at  a  discount  of  30^  and  5fo.  What  is  his 
gain  per  cent  ? 

9.  A  certain  publishing  house  allows  a  discount  of 
l(J|/o  on  all  orders  under  $100, 16|/o  and  10  fo  on  all  orders 
between  $100  and  $500,  and  16ffo,  10 fo  and  bfo  on  all 
orders  above  $  500.  If  three  dealers  wish  to  send  in  orders 
amounting  to  $60,  $175  and  $350  respectively,  how  much 
will  each  one  gain  if  they  combine  their  orders  ? 

10.  Which  is  the  better  discount  for  a  buyer  to  take : 

(a)  331/0,  10/0  and  bfo,  or  40/.  ? 

(5)  10/0,  10/0  and  Bfo,  or  25 fo  ? 

(c)  40/0  and  15/0,  or  40/),  lO/o  and  5fo? 

Id)  60 fo  and  15 fo,  or  60 fo? 

11.  How  much  above  the  cost  must  a  book  marked 
$2  be  sold,  if  10  fo  is  taken  from  the  marked  price  and 
a  profit  of  10  fo  on  the  cost  is  still  made  ? 


COMMERCIAL   DISCOUNTS  IGl 

12.  One  firm  offers  to  sell  -1500  wortli  of  galvanized 
pipe  at  a  clisconnt  of  40 J^,  10^  and  b'/o,  and  another  firm 
offers  a  discount  of  U\f,  20/o  and  lO/o.  Wliicli  is  the 
better  rate  of  discount  and  what  is  tlie  difference  in 
dollars  ? 

13.  Office  furniture  amounting  to  $  750  was  inventoried 
at  the  end  of  the  first  year  at  25 ^^  below  cost  and  at  the 
end  of  tlie  second  year  at  15  ^o  below  inventory.  What 
was  the  loss  in  value  ? 

14.  If  a  grocer  buys  sugar  at  3.42  ct.'per  pound  and 
sells  it  at  4  ct.,  what  is  his  gain  per  cent? 

15.  A  dealer  marked  his  goods  at  33^  J^  above  cost,  but 
sold  at  a  certain  per  cent  discount  and  still  made  15^  on 
the  cost.     What  was  the  rate  per  cent  of  discount  ? 

16.  What  three  equal  rates  of  discount  are  equivalent 
to  a  single  rate  of  27.1^? 


lyman's  adv.  ar. 11 


MARKING  GOODS 

240.  ]Most  merchants  use  a  private  mark  to  indicate  the 
cost  and  selling  price  of  goods.  They  usually  select  some 
word  or  phrase  containing  10  different  letters  and  use  it 
as  a  key.  These  letters  are  used  to  represent  the  9  digits 
and  0.  In  this  way  the  cost  and  selling  price  Avill  be 
understood  only  by  those  who  know  the  key. 

Two  different  keys  are  generally  selected,  one  to  mark  the  cost  and 
the  other  to  mark  the  selling  price.  One  or  more  extra  letters,  called 
repeaters,  are  used  to  avoid  the  repetition  of  a  figure  and  to  prevent 
giving  any  clew  to  the  private  mark  used.  The  cost  is  usually  written 
above  and  the  selling  price  below  a  line. 

241.  The  words  equinoctial  (omitting  the  last  i)  and 
importance  are  adapted  for  use  as  keys,  since  they  both 
contain  10  different  letters.  These  words  give  the  fol- 
lowing keys : 

1234567890 
e  q  u  i  n  0  c  t  a  I 
i   m  2?   0   r    t   a   71    c   e      Repeaters  x  and  ?/. 

Thus,  if  a  merchant  pays  $29.98  per  dozen  for  hats,  and  sells  them 
for  !$3.50  each,  he  would  mark  them  ^         • 

EXERCISE  55 

1.  Explain  why,  if  the  cost  of  a  dozen  articles  is 
divided  by  10,  the  result  will  give  the  retail  price  of  one 
article  with  a  profit  of  20%  added. 

162 


MARKING   GOODS  lf>3 

2.  Explain  wliy,  to  make  a  profit  of  33^%,  the  cost  of 
a  dozen  articles  may  be  divided  by  10  and  -1  of  the  result 
added. 

3.  Determine  sliort  methods  of  finding  the  retail  price 
of  one  article  when  the  cost  per  dozen  is  given  and  the 
dealer  wishes  to  make  a  profit  of  35%  ;  37|  % ;  40% ;  50%  ; 
60%. 

4.  A  merchant  buys  shirts  for  -$12.50  per  dozen.  For 
what  price  must  lie  sell  them  to  make  50%?    40%? 

5.  A  merchant  retails  neckties  at  50  ct.  and  makes  50% . 
How  much  did  they  cost  him  per  dozen  ? 

Using  equinoctal  and  importance  as  keys,  mark  the  cost 
and  selling  price  of  the  following  articles : 

6.  Gloves  costing  -$5  per  dozen  and  selling  for  $6.50. 

7.  Hats  costing  $22.50  per  dozen  and  selling  at  20% 
gain. 

8.  Caps  costing  $7.50  per  dozen  and  selling  at  33^% 
gain. 

9.  Shoes  costing  $1.98  and  selling  at  25%  gain. 

10.  Rubber  boots  costing  $2.68  and  selling  at  $3.75. 

11.  Make  a  key  of  the  letters  contained  in  the  words 
Cumberland  and  Charleston  spelled  backward,  and  mark 
the  articles  given  in  Ex.  6  to  10. 

12.  A  merchant  sold  a  bill  of  goods  that  cost  $125 ;  the 
asking  price  was  30%  in  advance  of  the  cost,  from  wliich 
a  wholesale  discount  of  15%  was  allowed.  What  was  the 
per  cent  gain  ? 

13.  An  invoice  of  hats  costing  $112  is  marked  so  as  to 
sell  at  40%  profit.  Does  the  merchant  gain  or  lose  if  the 
hats  are  sold  at  30%  discount  from  the  marked  price  ? 


COMMISSION   AND   BROKERAGE 

242.  Farmers,  produce  dealers,  manufacturers  and 
others  frequently  find  it  more  convenient  to  employ  a 
third  person  to  dispose  of  their  goods,  instead  of  selling 
direct  to  consumers.  The  person  who  sells  the  goods  is 
called  a  commission  merchant,  an  agent  or  a  broker.  The 
pay  received  for  such  services  is  called  commission  or 
brokerage. 

243.  Produce  is  usually  shipped  to  a  commission  mer- 
chant, and  sold  by  him  in  his  own  name.  The  proceeds 
less  the  commission,  or  the  net  proceeds,  are  sent  to  the 
shipper  or  consignor.  If  a  commission  merchant  is  buying 
goods  for  a  customer,  he  charges  the  cost  plus  the  com- 
mission. The  amount  of  commission  varies  in  different 
lines  of  business. 

244.  A  broker  buys  and  sells  without  having  possession 
of  the  goods,  and  generally  does  not  make  contracts  in  his 
own  name. 

245.  Commission,  or  brokerage,  is  usually  computed  at 
a  certain  per  cent  of  the  amount  realized  on  sales,  or  in- 
vested for  the  customer.  In  buying  and  selling  certain 
kinds  of  merchandise,  it  is  customary  to  pay  a  certain 
price  per  unit  of  measurement  or  weiglit;  as  grain  per 
bushel,  hay  per  ton,  etc. 

164 


COMMISSION  AND  BROKERAGE  165 

EXERCISE  56 
What  is  the  conimissiuu  uii : 

1.  1750.50  at  2%  ?  4.   1350.45  at  10%^ 

2.  il2368ati%?  5.   i|3764ati%? 

3.  875429.75  at  i%?  6.   i5250at7|%? 

7.  The  sale  of  1000  bii.  of  grain  at  |-  ct.  a  bushel  ? 

8.  The  sale  of  25  T.  of  hay  at  50  ct.  a  ton  ? 

9.  The  sale  of  40  liead  of  cattle  at  50  ct.  a  head  ? 

10.  The  sale  of  1500  bales  of  cotton  at  25  ct.  a  bale  ? 

11.  The  sale  of  22  horses  @  .$125  a  head  at  2%  ? 

Find  the  amount  to  invest  and  the  commission  when  the 
following  remittances  and  rates  of  commission  are  given : 

12.  $1030  at  3%.  15.   86300  at  5%. 

13.  15025  at  1%.  16.   f  1100  at  10%. 

14.  88020  at  i%.  17.   82562  at  21%. 

Find  the  net  proceeds  and  commission  on  each  of  the 
following  sales : 

18.  200  bbl.  of  apples  @  83,  less  freight  862.50,  com- 
mission 5%. 

19.  5000  bu.  of  wheat  @  72  ct.,  less  8102.50  freight, 
825  storage,  \%  insurance  and  2%  commission. 

*  20.    500  bbl.  of  beef  @  819.50,  less  48  ct.  a  barrel  freight, 
87.50  storage  and  2|%  commission. 

21.  1500  doz.  eggs  @  22  ct.,  less  89.50  express  and  10% 
commission. 

22.  12  bales  of  cotton  averaging  475  lb.  @  9^  ct.  a 
pound,  less  842.50  freight,  81.25  a  bale  storage  and  2^% 
commission. 


166  COMMISSION  AND  BROKERAGE 

23.  An  agent  charges  $20  for  advertising  the  sale  of  a 
farm,  and  3%  commission.  He  sells  the  farm  for  $7500. 
What  are  the  net  proceeds  and  the  agent's  commission  ? 

24.  A  collector  is  given  a  bill  of  $1350  to  collect  at  5% 
commission.  He  succeeds  in  collecting  85  ct.  on  the  dollar. 
How  much  is  due  his  employer,  and  what  is  his  commission  ? 

25.  A  miller  orders  his  agent  to  buy  him  2500  bu.  of 
Avheat  @  80  ct.  If  the  agent  charges  3%  commission,  and 
freight  and  drayage  charges  are  $95.75,  what  is  the  total 
cost  of  the  wheat  ? 

26.  A  merchant  sends  his  agent  $1836  to  buy  an  equal 
number  of  yards  of  each  of  three  grades  of  muslin  at 
3,  4  and  5  ct.  a  yard  respectively,  after  deducting  2% 
commission.  How  many  yards  of  each  kind  does  he  get, 
and  what  is  the  agent's  commission  ? 

27.  A  manufacturer  sold  $20000  worth  of  goods  through 
his  agent  at  2%  commission,  and  instructed  him  to  purchase 
raw  material  with  the  proceeds  at  1%  commission.  Find 
the  net  proceeds  of  the  sale,  the  amount  invested  in  raw 
material,  and  the  agent's  entire  commission. 

28.  A  dealer  sent  two  car  loads  of  hay  weighing  27  T. 
to  his  broker  in  New  York,  who  sold  it  for  $16  a  ton,  and 
remitted  $418.50.  If  the  dealer  paid  $8.50  a  ton,  the 
freight  cost  16|  ct.  a  hundredweight,  and  storage  was 
$12.50,  how  much  did  he  make,  and  what  was  the  broker's 
commission  per  ton  ? 

29.  A  book  agent  sells,  during  July  and  August,  77 
copies  of  a  certain  book  at  40%  commission.  If  he  sells 
20  copies  in  full  leather  binding  @$6.50,  25  copies  in  half 
leather  @  $5.25,  and  32  copies  in  cloth  @  $4,  how  much 
does  he  make  if  his  expenses  average  $1.25  a  day? 


INTEREST 

246.  Interest  is  money  paid  for  the  use  of  money. 

247.  Tlie  sum  loaned  is  called  the  principal. 

248.  Tlie  rate  of  interest  is  the  rate  per  cent  per  annum 
of  the  principal  paid  for  tlie  use  of  money. 

In  the  absence  of  a  specific  contract  the  rate  of  interest  is  fixed  by 
law  in  most  states.  The  rate  thus  determined  is  called  the  legal  rate. 
By  special  contract  interest  may  be  received  at  a  higher  rate  than  the 
legal  rate.  The  maximum  contract  rate  is  fixed  by  law  in  most  states. 
Interest  in  excess  of  the  maximum  contract  rate  is  called  usury.  The 
penalty  for  usury  is  fixed  by  law  in  states  where  it  is  forbidden. 

249.  The  principal  plus  the  interest  is  called  the  amount. 

250.  The  practical  business  problem  of  most  frequent 
occurrence  in  interest  is  to  find  the  interest  when  the  prin- 
cipal^ rate^  and  time  are  given. 

251.  In  computing  interest  without  tables  it  is  usually 
tlie  custom  to  reckon  the  year  as  360  da.,  the  month  as 
.^^2  of  ^  year  and  the  day  as  -^-^  of  a  month  or  ^i^  of  a  year. 
(See,  however,  §  254.) 

Ex.  Find  the  interest  and  amount  of  $720  for  2  yr. 
6  mo.  15  da.  at  6^. 

Solution.     2  yr.  6  mo.  15  da.  =  f|f  yr. 
The  interest  for  one  year  =  0.06  of  ^720. 
.-.  the  interest  for  f|A  yr.  =  fi§  x  ^^^  x  $720  =  $109.80, 
The  amount  is  |720  -f- 1109.80  =  $829.80. 

167 


168  INTEREST 

EXERCISE  57 

Find  the  interest  and  amount  of : 

1.  $100  for  1  yr.  4  mo.  at  Q% 

2.  $125  for  6  yr.  1  mo.  20  da.  at  If. 

3.  $150  for  5  yr.  9  mo.  11  da.  at  If. 

4.  $50  for  4  yr.  11  mo.  10  da.  at  {jf. 

5.  $1000  for  5  mo.  10  da.  at  If. 

6.  $350  for  3  yr.  9  mo.  at  If. 

7.  $1500  for  2  mo.  at  6/o. 

8.  $25  for  1  yr.  at  %f. 

9.  $1200  for  2  yr.  6  mo.  at  bf. 

10.    $500  for  2  yr.  3  mo.  15  da.  at  -if. 

252.  A  short  ynethod  of  computing  interest  at  Qf  is  based 
on  the  year  of  12  mo.  of  30  da.  each.  This  method  is 
sometimes  called  the  ^f  method. 

The  interest  on  ,|1  for  1  yr.  at  6%  =  $0.06. 

The  interest  on  %l  lor  1  mo.  at  6%  =  J^  of  ^^-^^  =  1 0.005. 
The  interest  on  $1  for  1  da.  at  6%  =  gV  of  $0,005  =  $0.0001. 

Ex.    Find  the  interest  on  $250  for  2  yr.  4  mo.  12  da. 

at  Qf. 

The  interest  on  $  1  for  2  yr.  at  6%=  2  x  $0.06  =  $0.12. 

The  interest  on  $1  for  4  nio.  at  6%  =  4  x  $0,005      =  $0.02. 

The  interest  on  $1  for  12  da.  at  6%  =  12  x  $0.000i  =$0.002. 

.-.  the  interest  on  $1  for  2  yr.  4  mo.  12  da.  at  6%     =  $0,142. 

.-.  the  interest  on  $250  for  2  yr.  4  mo.  12  da.  =  250  x  $0,142  =  $35.50. 

253.  To  find  the  interest  at  bf,  subtract  J-  of  the  interest 
at  6  %  ;  at  7  %,  add  1  of  the  interest  at  6  %,  etc. 


SIMPLE  INTEREST  109 

EXERCISE    58 

1.  By  the  6%  nictliod  find  llie  interest  on  ^r^lOO  for 
2  yr.  3  mo.  10  da.  at  4  %  ;  at  4|  %  ;  at  «Ji  %  ;  at  7  ^  %  ;  at 
3%  ;   at  31%. 

Find  the  interest  on  : 

2.  $325  for  1  yr.  2  mo.  at  6%. 

3.  $450  for  2  yr.  3  mo.  14  da.  at  5,]  %. 

4.  #315.T5  for  2  mo.  15  da.  at  8%. 

5.  12000  for  30  da.  at  6%. 

6.  $115.50  for  3  mo.  10  da.  at  1},%. 

7.  $387.50  for  6  mo.  at  5%. 

8.  $524.70  for  60  da.  at  (5%. 

9.  $97.30  for  3  mo.  10  da.  at  7  %. 

10.    $80.60  for  1  yr.  6  mo.  15  da.  at  3%. 

254.  Exact  Interest.  To  find  the  exact  interest  we  must 
take  the  exact  number  of  days  between  dates  and  reckon 
365  da.  to  a  year.  Exact  interest  is  used  by  the  United 
States  government,  by  some  banks,  and  to  some  extent  in 
other  business  transactions. 

Ux.  Find  the  exact  interest  on  $2500  from  April  10  to 
Sept.  5  at  5%. 

148  =  the  number  of  days  from  April  10  to  Sept.  5. 

.-.  the  interest  on  |2500  for  118  da.  at  50/  -  iliiii§2ifM^=:  i$50.68. 

^'  100  X  305 


170  INTEREST 

EXERCISE   59 

Find  the  exact  interest  on  : 

1.  1575  from  July  5  to  Sept.  5  at  7  %. 

2.  1125  from  Jan.  1  till  Nov.  1  at  6  %. 

3.  110000  from  March  10  till  June  1  at  5%, 

4.  1375.30  from  April  25  till  Aug.  1  at  6%. 

5.  Find  the  amount  of  ^55  375  at  G%  exact  interest  from 
Nov.  11,  1903,  till  July  27,  1905. 

6.  May  10,  1903,  -$500  is  loaned  at  6%.  Find  the 
amount  due  Sept.  1,  1905,  exact  interest. 

7.  If  1500  is  loaned  on  July  28,  1905,  when  will  it 
amount  to  8720  ? 

8.  What  is  the  difference  between  the  exact  interest 
and  the  common  interest  on  81000  from  July  1  till  Nov.  1 
at  6%?  If  the  exact  number  of  days  between  dates  and 
360  days  to  the  year  are  taken,  how  much  does  the  com- 
mon interest  differ  from  the  exact  interest  ? 

9.  Show  that  the  difference  between  the  common  inter- 
est and  the  exact  interest  is  ^^  of  the  former  and  ^^  of  the 
latter. 

10.  Hence,  show  that  exact  interest  may  be  obtained  by 
subtracting  y^g  part  from  the  conuiion  interest,  and  the 
common  interest  may  be  obtained  from  the  exact  interest 
by  adding  ^^  part  of  itself. 

Exact  interest  is  the  fairest,  but  on  account  of  its  incon- 
venience without  tables  is  not  generally  used. 

255.  The  following  is  a  section  of  an  interest  table  for 
the  year  of  365  da.  at  6  %  : 


SIMPLE  INTEREST 


171 


l).v\> 

iii..,i 

■_'(M)0 

.SODI) 

4(iUit 

.".(((III 

i;(i  III 

Tniin 

>iiiiii  j 

60 
61 
62 
63 
64 
65 

m 

67 
68 
69 
70 
71 
72 
73 
74 
75 
76 
77 
78 
79 
80 

9.863 
10.027 
10.192 
10.356 
10.521 
10.685 
10.849 
11.014 
11.178 
11.342 
11.507 
11.671 
11.836 
12.000 
12.164 
12.329 
12.493 
12.658 
12.822 
12.986 
13.151 

19.726 
20.055 
20.384 
20.712 
21.041 
21.370 
21.699 
22.027 
22.356 
22.685 
23.014 
23.342 
23.671 
24.000 
24.329 
24.658 
24.986 
25.315 
25.644 
25.973 
26.301 

29.589 
30.082 
30.575 
31.068 
31.562 
32.055 
32.548 
33.041 
33.534 
34.027 
34.521 
35.014 
35. 507 
36.000 
36.493 
36.986 
37.479 
37.973 
38.466 
39.959 
39.452 

39.452 
40.110 
40.767 
41.425 
42.082 
42.740 
43.397 
44.055 
44.712 
45.370 
46.027 
46.685 
47.342 
48.000 
48.658 
49.315 
49.973 
50.630 
51.288 
51.945 
52.603 

49.315 
50.137 
50.959 
51.781 
52.603 
53.415 
54.247 
55.068 
55.890 
56.712 
57.534 
58.356 
59.178 
60.000 
60.822 
61.644 
62.466 
63.288 
64.110 
64.932 
65.753 

59.178 
60.164 
61.151 
62.137 
63.123 
64.110 
65.09(5 
66.082 
67.068 
68.055 
69.041 
70.027 
71.014 
72.000 
72.986 
73.973 
74.959 
75.945 
76.932 
77.918 
78.904 

69.041 
70.192 
71.342 
72.493 
73.644 
74.795 
75.945 
77.096 
78.947 
79.397 
80.548 
81.699 
82.849 
84.000 
85.151 
86.301 
87.452 
88.603 
89.753 
90.904 
92.055 

78.904 
80.219 
81.534 
82.849 
84.1(J4 
85.479 
86.795 
88.110 
89.425 
90.740 
92.055 
93.370 
94.685 
96.000 
97.315 
98.630 
99.945 
101.260 
102.575 
103.890 
105.205 

88.7(J7 

90.247 

91.726 

93.205 

94.685 

96.164 

97.644 

99.123 

100.603 

103.562 

103.562 

105.041 

106.521 

108.000 

109.479 

110.959 

112.438 

113.918 

115.397 

116.877 

118.356 

Years 

1000 

2000 

3000 

4000 

5000 

6000 

7000 

8000 

9000 

1 
2 

3 
4 

5 
■  6 

60 
120 
180 
240 
300 
360 

120 
240 
360 

480 
600 
720 

180 
360 
540 
720 
900 
1080 

240 

480 

720 

960 

1200 

1440 

300 

600 

900 

1200 

1500 

1800 

360 
720 
1080 
1440 
1800 
2160 

420 
840 
1260 
1680 
2100 
2520 

480 
960 
1440 
1920 
2400 
2880 

510 
1080 
1620 
2160 
2700 
3240 

Ux.     By  the  use  of  the  table  find  the  interest  on  -^4650 

for2yr.   67  da.  at  6 /o.  for  2  yr.  for  67  da. 

The  interest  on  .'$4000  =  $480  +  $44.06 
The  interest  on  600  =  72  +  6.61 
The  interest  on  50  =  6  +  0.55 
The  interest  on  $4650  =  $558  +  $51.22  =  $609.22. 


172  INTEREST 

EXERCISE   60 
By  the  use  of  the  table  find  the  interest  on : 

1.  1500  for  {^^  da. 

2.  11000  for  60  da. 

3.  15225  for  73  da. 

4.  110575  for  1  yr.  60  da. 

5.  $1846  for  2  yr.  80  da. 

6.  11710  for  75  da. 

7.  il250  for  63  da. 

8.  12120  from  July  2  till  Sept.  5. 

9.  1648.60  from  Jan.  10  till  March  15. 
10.   -11410  from  May  1  till  July  10. 

256.  In  any  problem  in  interest  there  are  four  elements 
involved,  tlie  principal,  the  rate,  the  time  and  the  interest. 
When  any  three  of  tliese  are  given,  the  other  can  be  found. 
As  indicated  above,  the  practical  business  problem  is  to 
find  the  interest  when  the  principal,  rate  and  the  time  are 
given.  However,  the  principles  involved  in  the  following 
illustrative  examples  are  frequently  met  with  in  business  : 

Ex.  1.  What  principal  will  produce  f  72  interest  in  1  yr. 
6mo.  at6fo? 

Solution.     Let  x  =  the  principal. 

Then  «  x  (j%oi  x  =  ^  72. 

.-.  x  =  ~^^'^^  =$800. 
3  X  0.06 


SIMPLE   IXTEHEST  173 

E.r.  2.    At  what  rate  will  6^800  produce  872  interest  in 
2  yrs.  ? 

Solution.     Let  a:  %=  the  rate. 

Then  .r  %  x  2  of  ^  800  =  8  72. 

•       7-  0/    —  ^    '  -  —       9      _   4. 1  0/ 

E.V.  3.    In  what  time  will -S 1000  produce  >=70  interest  at 

Solution.     Let  .r  =  the  time. 

Then  x  x  4%  of  8  1000  =  870. 

S70 
0.04x81000      '■ 

.-.  .r  =  1  yr.,  or  1  yr.  9  mo. 

Ex.  4.    What  prinei})al  will  amount  to  81238  in  6  mo. 
10  da.  at  6  ^  ? 

Solution.     Let  .r  =  the  principal. 

Then  x  +  6  %  x  J§§  of  x  =  8  12:]8  =  the  amount. 

.-.  x  = ^^-^^^ =  81200. 

1  +  0.0(5  X  ie 

EXERCISE   61 

Find  the  rate  at  which  : 

1.  8750  will  produce  867.50  interest  in  1  yr.  6  mo. 

2.  82000  will  produce  8105  interest  in  9  mo. 

Find  the  time  in  which  : 

3.  8250  will  produce  825  interest  at  5^. 

4.  81200  will  produce  890  interest  at  6f). 


9 
9 


174  INTEREST 

5.  1850  will  produce  -f  106.25  at  5/o. 

6.  $2000  will  produce  $105  at  If. 

What  principal  will  produce  : 

7.  il08  interest  in  1  yr.  6  mo.  at  6^  ^ 

8.  $61.25  interest  in  2  yr.  6  mo.  at  7^ 

9.  $262.50  interest  in  1  yr.  6  mo.  at  5%  ? 

What  principal  will  amount  to : 

10.  $575  in  2  yr.  6  mo.  at  Qf  ? 

11.  $1050  in  1  yr.  at  5/o? 

12.  $1570  in  1  yr.  2  mo.  at  4fo  ? 

13.  A  man  with  $25000  invested  in  his  business  makes 
121^  profit  annually.  He  sells  out  and  invests  the  $25000 
at  6fo  and  works  on  a  salary  of  $2000  per  annum.  Does 
he  make  or  lose  by  the  change  and  how  much  ? 

14.  A  man  invests  $20000  in  business  and  makes  $6000 
in  one  j^ear  on  his  sales.  If  the  total  expenses  of  running 
the  business  are  $  3500,  what  rate  does  he  make  on  his 
money  ? 

15.  A  house  and  lot  costs  $1800  and  rents  for  $16  a 
month.  If  taxes,  insurance  and  repairs  cost  $72  a  year, 
what  rate  is  earned  on  the  investment  ? 

16.  Jan.  1,  1900,  $450  are  deposited  in  a  savings  bank 
at  3/o.     Find  the  amount  due  July  3,  1900. 

257.  Compound  Interest.  In  compound  interest  the 
interest   is    added   to   the   principal    at  the   end   of   each 


COMPOUND  INTEREST  11 0 

interest  period.     Then  the  amount  becomes  the  new  prin- 
cipal for  the  next  interest  period. 

Unless  otherwise  stated,  interest  is  compounded  annually,  though 
it  may  be  compounded  semiannually,  (juarteriy,  etc.,  l)y  agreement. 
In  most  states  compound  interest  cannot  be  collected  by  law,  but  pay- 
ment of  it  does  not  constitute  usury. 

Ux.  Find  the  compound  interest  on  8500  for  3  yr. 
4  mo.  15  da.  at  4%. 

Solution.     $500  =  principal  first  year. 
0.04 


20.00  =  interest  first  year 
500 


$520.00  =  amount  first  year  =  principal  second  year. 
0.04 


20.80  =  interest  second  year. 
520 


$540.80  =  amount  second  year  =  principal  third  year. 
0.04 


21.63  =  interest  third  year. 
540.80 


$562.43  =  amount  third  year  =  principal  fourth  year. 

Interest  on  $562.43  for  4  mo.  15  da.  at  4%  =  88.44. 

$562.43  +  $8.44  =  $570.87  =  amount  for  3  yr.  4  mo.  15  da. 
500 


$70.87  =  compound  interest  for  3  yr.  4  mo.  15  da. 

258.  The  chief  use  of  compound  interest  is  among  large 
investors,  such  as  life  insurance  companies,  building  and 
loan  associations,  private  banking  establishments,  etc., 
who  wish  to  compute  the  income  from  reinvestment  of 
interest  when  due.  For  such  work  compound  interest 
tables  are  used. 


176 


INTEREST 


Tlie  following  is  a  section  of  such  a  table : 


Periods 

1  Per  Cent 

1^  Per  Cent 

2  Per  Cent 

3  Per  Cent 

4  Per  Cent 

1 

1.0100000 

1.015000 

1.020000 

1.030000 

1.040000 

2 

1.0201000 

1.030225 

1.040400 

1.060900 

1.081600 

3 

1.0303010 

1.045678 

1.061208 

1.092727 

1.124864 

4 

1. 0406040 

1.061364 

1.082432 

1.125509 

1.169859 

5 

1.0510100 

1.077284 

1.104081 

1.159274 

1.216653 

6 

1.^615201 

1.093443 

1.126162 

1.194052 

1.265319 

7 

1.0721353 

1.109845 

1.148686 

1.229874 

1.315932 

8 

1.0828567 

1.126493 

1.171660 

1.266770 

1.368569 

9 

1.0936852 

1.143390 

1.195093 

1.304773 

1.42.3312 

10 

1.1046221 

1.160541 

1.218994 

1.343916 

1.480244 

11 

1.1156683 

1.177949 

1.243374 

1.384234 

1.539454 

12 

1.1268250 

1.195618 

1.268242 

1.425761 

1.601032 

13 

1.1380'.)32 

1.21.3552 

1.293607 

1.468534 

1.665074 

14 

1.1494742 

1.231756 

1.319479 

1.512590 

1.731676 

15 

1.1609689 

1.250232 

1.345868 

1.557967 

1.800944 

16 

1.1725786 

1.268985 

1.372786 

1.604706 

1.872981 

17 

1.1843044 

1.288020 

1.400241 

1.652848 

1.947901 

18 

1.1961474 

1.307341 

1.428246 

1.702433 

2.025817 

19 

1.2081089 

1.326951 

1.456811 

1.753506 

2.106849 

20 

1.2201900 

1.346855 

1.485947 

1.806111 

2.191123 

Solution  of  the  above  example  by  means  of  the  tables. 
The  amount  of  -|1  for  3  yr.  at  4%  is  S^  1.12486. 
The  amount  of  -|500  will  be  500  x  -11.12486  =  $562.43. 
The  example  may  now  be  completed  by  using  the  tables  for  simple 
interest  for  4  mo.  15  da.,  or  as  on  p.  175. 


EXERCISE  62 

1.  Find  the  compound  amount  and  the  compound 
interest  of  12000  for  3  yr.  6  mo.  at  4%  payable  semi- 
annually. 

Note.  It  is  evident  that  if  interest  is  4  %  compounded  semiannnally 
for  3  yr.  6  mo.,  the  amount  is  the  same  as  if  the  rate  is  2%  com- 
pounded annually  for  7  yr. 


ANNUAL    INTEREST  177 

2.  What  is  tlie  difference  between  the  simple  iind  com- 
pound interest  on  $750  for  2  yr.  7  mo.  iit  5%? 

3.  Find  tlie  amount  of  ifSOOO  compounded  annually  for 
4  yr.  at  4%. 

4.  Find  tlie  amount  of  88500  compounded  semiannually 
for  5  yr.  at  8%  ;   at  4%  ;   at  (>%. 

259.  Annual  Interest.  If  a  note  or  other  written  agree- 
ment contains  the  expression  "with  annual  interest"'  or 
'^  with  interest  payable  annually,"  the  interest  is  due  at 
the  end  of  each  year,  and  if  not  then  paid,  will  draw  simple 
interest  until  paid.  Such  a  note  or  agreement  is  said  to 
bear  annual  interest. 

As  in  the  case  of  compound  interest,  in  most  states  annual  interest 
cannot  be  collected  by  law,  but  does  not  constitute  usury. 

Ux.  George  Reed  borrowed  $1500  at  7%,  and  agreed 
to  pay  interest  annually.  Having  paid  no  interest,  he 
wishes  to  settle  at  the  end  of  3  yr.  3  mo.  20  da.  What 
is  the  amount  due  ? 

The  simple  interest  on  $  1500  for  3  yr.  3  mo.  20  da.  at  7%  =  8347.08. 
Then,  in  addition  to  this,  the  simple  interest  on 

1105  at  7%  for  2  yr.  3  mo.  20  da. 
$105  at  7%  for  1  yr.  3  mo.  20  da. 
$105  at  7%  for  3  mo.  20  da. 

or  on  $105  at  7  %  for  3  yr.  10  mo.  =  |28.18. 

Hence,  principal  borrowed  =  $1500 

Simple  interest  =       3-47.08 

Simple  interest  on  interest  not  paid  when  due  =         28.18 
.-.  total  amount  due  at  annual  interest  =  $1875.26 

lyman's  adv.  ar.  — 12 


178  INTEREST 

EXERCISE   63 

1.  What  is  the  difference  between  the  simple  interest 
and  the  annual  interest  in  the  preceding  example  ?  How 
long  is  it  after  the  date  on  which  the  money  is  borrowed 
before  the  annual  interest  begins  to  differ  from  the  simple 
interest  ? 

2.  What  is  the  difference  between  the  compound  in- 
terest and  the  annual  interest  in  the  preceding  example  ? 
How  long  is  it  before  the  compound  interest  begins  to 
differ  from  tlie  simple  interest?  from  the  annual  interest? 

3.  Find  what  f  2500  will  amount  to  in  4  yr.  10  mo. 
18  da.  at  5%  simple  interest  and  at  5%  compound  interest. 

4.  Sept.  1,  1896,  a  man  borrows  $500  at  6%  interest, 
payable  annually.  If  nothing  is  paid  until  Dec.  1,  1901, 
how  much  is  due  ? 

5.  Notes  are  sometimes  given  with  interest  coupons 
attached.  These  coupons  draw  interest,  frequently  at  a 
higher  rate  than  the  note  itself,  if  not  paid  when  due.  A 
coupon  note  for  12200  is  issued  July  1,  1896,  at  6%  inter- 
est. Nothing  is  paid  until  July  1, 1902.  Find  the  amount 
at  that  date,  the  coupons  bearing  7%  if  not  paid  when  due. 

260.  Promissory  Notes.  A  promissory  note  is  a  written 
promise  to  pay  to  a  certain  person  named  in  the  note  a 
specified  sum  of  money  on  demand,  or  at  a  specified  time. 

261.  A  promissory  note  is  negotiable,  i.e.  can  be  trans- 
ferred from  one  owner  to  another  by  indorsement  when 
it  is  made  payable  to  the  order  of  a  definite  person,  or  to 
hearer. 


PEOMISSORY  NOTES 


179 


The    following    is    a    coininon    form    of    a    negotiable 
promissory  note  : 


//^^^. 

^etvait, 

muk.,  nux^k 

,  27,  f^O^. 

c/t^tl/ 

daya^ 

after  date 

I 

promise  to 

])ay  to  the 

order  of  /f&tcvif 

joi^fi&Qy  an& 

tko-^l^(^'}^cL  dncL 

^  dotU^a., 

lUO                       ' 

value  received, 

^v~itk  vnt&v&Qyt 

at  6%. 

Mo.  ^3. 

Due 

CCnclv&ii^  ^aknoyayi. 

Andrew  Johnson  is  the  maker  of  this  note,  Henry 
James  is  the  payee,  and  $1000  is  the  face. 

262.  The  above  note  would  be  non  negotiable  if  the 
words  ''the  order  of"  were  omitted.  In  that  case  the 
note  would  be  payable  to  Henry  James  only. 

263.  If  a  note  is  sold  by  the  payee,  he  must  indorse  it 
by  w^riting  his  name  across  the  back. 

264.  There  are  three  common  forms  of  indorsement : 

(1)  In  blank,  the  indorser 
simply  writing  his  name  across 
the  back,  thus  making  the  note 
payable  to  the  bearer. 


Indorsement   in   Blank 


(2)  In  full,  the  indorser  di- 
recting the  payment  to  the 
order  of  a  definite  person. 


Indorsement  in  Full 

c^a-2^  ta  tk&  avd&v  aj^ 
^viy-l&u  CLnct  f'foJjiyk'. 


180 


INTEREST 


(3)  Qualified,  the  indorser 
avoiding  responsibility  by  writ- 
ing the  words  ''  without  re- 
course "  over  his  name. 


Qualified  Indorsement 

S^ciu  ta  tk&  avcC&v  o-i 
ofvlf-l&i^  cincL  ffat^k. 
lAMVio-iit  V£.(S^auv^&. 

OR    SIMPLY 


265.  By  indorsing  a  note  either  in  blank  or  in  full,  the 
payee  becomes  responsible  for  its  payment  if  the  maker 
fails  to  pay  it.  The  indorsement  in  full  will  insure  greater 
safety  since  in  this  case  the  note  is  made  payable  to  a 
definite  person. 

266.  A  note  made  payable  to  Henry  James,  or  bearer, 
is  also  negotiable,  but  does  not  need  indorsement. 

267.  The  custom  of  allowing  three  days  of  grace  in  the 
payment  of  a  note  has  been  abolished  in  many  states  and 
is  rarely  used  in  others. 

268.  The  note  ou  pag'e  179  will  mature  March  27  +  60  days,  or 
May  26,  if  no  grace  is  allowed.  It  will  mature  March  27  +  63  days, 
or  May  29,  if  grace  is  allowed.  In  states  where  grace  is  allowed  this 
is  indicated  by  writing  in  the  note,  "  Due  May  26/29." 

269.  In  most  states  a  note  falling  due  on  Sunday  or  a  legal  holiday 
must  be  paid  the  preceding  business  day. 


EXERCISE   64 


1.  Write  a  30-day  promivssory  note  for  i500,  payable  to 
the  order  of  James  Black,  bearing  the  legal  rate  of  interest 
in  your  state.  By  indorsement  make  the  note  payable  to 
Henry  Wood. 


PARTIAL    PAYMENTS  181 

2.  What  is  the  maximnni  contract  rate  of  interest  in 
your  state  ?     What  is  the  penalty  for  usury  ? 

3.  Write  a  60-day  promissory  note  for  f  100,  with  your- 
self as  maker  and  Charles  Jennings  as  payee,  the  note 
bearing  the  date  May  5,  1903.  If  the  note  is  payable  to 
Charles  Jennings,  or  bearer,  in  what  way  may  it  be  trans- 
ferred ?  If  made  payable  to  Charles  Jennings,  or  order, 
in  what  way  may  it  be  transferred  ? 

4.  Find  the  date  of  maturity  of  the  note  required  in 
Ex.  3.     Add  days  of  grace  if  they  are  used  in  your  state. 

5.  Find  the  interest  on  the  above  note. 

6.  1250.  Ypsilanti,  Mich.,  April  11,  1905. 

Ninety  days  after  date  I  promise  to  pay  to  the 
order  of  William  Jordan  tw^o  hundred  fifty  and 
y^U^Q  dollars,  value  received,  with  interest  at  5%. 

L.  M.  Davis. 

When  is  the  above  note  due  ?  Wliat  is  the  amount 
at  maturity  ?  Who  pays  the  note  ?  Who  receives  the 
money?  Who  receives  the  note  when  paid?  What 
indorsement  is  necessary  if  the  note  is  sold  to  John 
Brown  ? 

270.  Partial  Payments.  Frequently  the  maker  of  a 
note,  not  being  able  to  pay  the  whole  amount  at  one  time, 
makes  several  partial  payments,  which  are  indorsed  on  the 
back  of  the  note  with  the  date  of  payment. 

Ux.  April  6,  1902,  a  man  buys  a  farm  for  $7500,  pay- 
ing 15000  in  cash,  and  giving  a  note  for  the  remainder  at 
6%,  with  the  privilege  of  paying  all  or  part  of  it  any  time 


182  INTEREST 

within  3  yr.      The  following  payments   are    made   and 
indorsed  on  the  note  by  the  payee : 

Oct.  1,  1903,  $  500 
April  1,  1904,  I  50 
Oct.    1,  1901,  -^1000 

What  amount  is  due  April  6,  1905  ? 
Face  of  note =  |2500.00 


yr. 

mo. 

da. 

Oct.    1,1903  =  1903 

10 

1 

April  6,  1902  =  1902 

4 

6 

The  interest  on  |2500  for  1  yr.  5  mo.  25  da.  at  6%  =  |  222.92 


Amount  due  Oct.  1,  1903  . 

. 

=    2722.92 

Payment 

. 

=      500.00 

Balance  due  =  new  principal     . 

• 

=  12222.92 

yv. 

mo. 

da. 

Oct.  1,  1904  =  1904 

10 

1 

Oct.  1,  1903  =  1903 

10 

1 

1 

The  interest  on  $2222.92  for  1  yr.  at  G%   .         .  =  8   133.38 

Amount  due  Oct.  1,  1904 =    2356.30 

Payment  April  1,  1904  (less  than  interest  due 

April  1,  viz.  $60.69) =        50.00 

Payment  Oct.  1,  1904 =     1000.00 

Balance  due  =  new  principal    .         .         .         .  =  $1306.30 

yr.  mo.       da. 

April  6,  1905  =  1905        4      6 
Oct.    1,  1904  =  1904       10       1 


6       5 
The  interest  on  $1306.30  for  6  mo.  5  da.   .         .     =        40.28 


.-.  the  amount  due  April  6,  1905,  is  .         .         .     =$1346.58 


PARTIAL   PAYMENTS  183 


Check  Difference  between  Dates 

Partial  I>ikfekence8 
yr.  mo.      da.  yr.       mo.       da. 

Date  of  settlement     1905       4       6  1         5      25 

Date  of  note  1902       4       6  1 


Difference  in  time  =3  6 


2       11       80  =  ::;yr. 

271.  Tlie  above  exami)le  is  solved  by  the  United  States 
Rule  of  Partial  Payments,  which  is  the  legal  method  in 
most  states.  i>y  this  method  the  amount  of  the  note  is 
found  to  the  time  Avhen  the  payment,  or  the  sum  of  the 
payments,  equals  or  exceeds  tlie  interest  due.  From  this 
amount  the  payment,  or  sum  of  the  payments,  is  sub- 
tracted. This  operation  is  repeated  to  the  time  of  the 
next  payment  and  so  on. 

272.  It  will  be  seen  that  three  cases  may  arise  under 
this  rule  : 

(1)  The  payment  may  be  exactly  equal  to  the  interest  due.  In 
this  case  the  payment  simply  cancels  the  interest,  and  the  balance  due 
remains  the  same  as  the  original  principal. 

(2)  The  payment  maiy  be  greater  than  the  interest  due.  In  this 
case  the  balance  due  is  diminished  by  the  amount  the  payment 
exceeds  the  interest  due. 

(3)  The  payment  may  be  less  than  the  interest  due.  In  this  case 
if^the  unpaid  balance  of  the  interest  were  added,  the  principal  would 
be  increased,  and  the  debtor  would  be  paying  more  interest  than  if  he 
had  made  no  payment  at  all.  Therefore,  when  the  payment  is  less 
than  the  interest  due,  no  change  is  made  at  that  time  in  the  principal ; 
but  the  interest  is  reckoned  to  the  date  when  the  sum  of  the  payments 
does  exceed  the  interest  due,  and  then  the  sum  of  these  payments  is 
subtracted. 

273.  The  following  metliod  of  solving  problems  in 
partial   payments,  called   The   Merchants*    Rule,    is   used 


184  INTEREST 

among  many  business  men  when  the  note  runs  for  one 
year  or  less : 

Ex.  Sept.  1,  1904,  a  merchant  takes  a  note  for  $827.50 
from  a  customer  in  payment  for  some  goods.  The  note  is 
to  run  for  1  yr.  at  6%.  During  the  year  the  following 
payments  are  made:  Nov.  1,  1904,  !^75;  April  1,  1905, 
1 100;  Aug.  1,  1905,  850.  Find  tlie  amount  due  Sept.  1, 
1905. 

The  amount  of  1327.50  for  1  yr.  at  6%  =  $347.15 

The  amount  of  -"^75  for  10  mo.  at  6%      =  ^  78.75 

The  amount  of  1100  for  5  mo.  at  (')%      =    102.50 

The  amount  of  |50  for  1  mo.  at  0%        =      .50.25  231..50 

Balance  due  Sept.  1,  1905  i|115.65 

274.  By  this  method  the  amount  of  the  note  is  found  from  the 
date  of  the  note  to  the  time  of  settlement.  The  amount  of  each 
payment  is  also  found  from  its  date  to  the  time  of  settlement.  The 
sum  of  the  amounts  of  the  payments  is  then  subtracted  from  the 
amount  of  the  principal. 

275.  Some  states,  e.g.  New  Hampshire  and  Vermont, 
have  rules  of  their  own  for  solving  problems  in  partial 
payments.  In  such  states  it  is  left  for  the  teacher  to 
present  the  rule. 

EXERCISE    65 

1.  Which  of  the  above  methods  is  better  for  the  debtor? 
Which  is  better  for  the  creditor  ? 

2.  $325  Cleveland,  Ohio,  May  15,  1897. 

Three  years  after  date  I  promise  to  pay  W.  W. 
Johnson,  or  order,  three  hundred  twenty-five  dollars, 
value  received,  with  interest  at  6%. 

Henry  George. 
Indorsements:  May  15,  1897,  122.75;  May  15,  1898, 
822.75;   June  29,  1900,  $100  ;  Jnne  12,  1902,  f  50.     Find 
the  amount  due  June  12,  1904. 


PARTIAL   PAYMENTS  185 

3.  Jan.  30, 1897,  Arthur  Ross  borrowed  1 150  ;  May  10, 
1897,  1125;  and  Dec.  10,  1902,  flOO,  all  at  7%  interest. 
He  paid  Oct.  1,  1901,  -flOO;  and  Dec.  10,  1902,  .$100. 
Find  tlie  amount  due  Marcli  10,  190^3. 

4.  March  1,  1897,  a  man  buys  a  farm  for  $5  6000.  He 
pays  $3000  in  cash  and  gives  a  note  for  the  remainder. 
He  makes  a  payment  of  $500  on  each  of  the  following- 
dates  :  March  1,  1898  ;  March  1,  1900  ;  Sept.  1,  1900  ;  and 
March  1,  1901.     Find  the  amount  due  March  1,  1902. 

5.  13500.  Ypsilanti,  Mich.,  Aug.  15,  1899. 

Five  years  after  date  T  promise  to  pay  John 
Robinson,  or  order,  three  thousand  five  hundred 
dollars,  value  received,  with  interest  at  G%. 

James  Rowe. 

Indorsements:  Nov.  5,  1902,  $300;  March  14,  1904, 
$200;  May  14,  1905,  $2000.  Find  the  amount  due 
Aug.  14,  1905. 

6.  A  note  for  $5000,  dated  March  1,  1903,  and  payable 
two  years  from  date,  with  interest  at  5%,  has  on  it  the 
following  indorsements :    April  1,  1903,  $500 ;    June    1, 

1903,  $500;  Sept.  1,  1903,  $200;  and  May  1,  1904,  $500. 
Find  the  amount  due  March  1,  1905. 

7.  Dec.  10,  1903,  a  merchant  takes  a  note  for  $260  to 
run  1  yr.  at  7%.  During  tlie  year  the  following  payments 
are  made:  Feb.  1, 1904,  $50;  June  21, 1904,  $25;   Oct.  10, 

1904,  $100.     Find  the  amount  due  Dec.  10,  1904. 


BANKS   AND   BANKING 

276.  A  bank  is  an  institution  that  deals  in  money  and 
credit.  Credit  is  a  promise  to  pay  money  in  tlie  future. 
Tlie  chief  instruments  of  credit  are  checks,  drafts,  and 
notes. 

It  is  a  mistake  to  say  that  banks  deal  only  in  money.  Their  most 
profitable  business  is  in  credit  transactions.  Banks,  however,  have  a 
capital  of  their  own  w-hich  serves  as  a  guarantee  fund.  Neither  do 
all  bank  deposits  represent  money  intrusted  to  banks  by  individuals. 
Most  of  them  represent  credit  loaned  to  individuals  by  banks.  Thus, 
if  a  bank  accepts  a  promissory  note  for  $5000,  it  may  give  in  return 
a  deposit  credit  for  |5000  less  the  discount  and  will  thereby  add  that 
sum  to  its  deposits. 

277.  There  are  several  kinds  of  banks,  among  which 
are  national  banks,  organized  under  the  National  Banking 
Act  of  186-3  and  the  amendments  that  have  been  made 
thereto  ;  state  banks,  organized  under  the  laws  of  the 
state  in  Avhich  they  are  situated  ;  savings  banks ;  and 
private  banks. 

278.  National  banks  are  subject  to  rigorous  supervision 
by  federal  authorities.  All  banks  organized  under  state 
laws  are  subject  to  similar  supervision  by  state  authorities. 

279.  The  chief  functions  of  banks  are  to  receive  deposits, 
to  lend  money  on  promissory  notes,  bonds,  and  mortgages, 
to  discount  merchants'  notes  befoi-e  they  are  due,  and  to 
buy,  sell,  and  collect  drafts  or  bills  of  exchange.     National 

186 


BANKS  AND  BANKING  187 

banks  also  issue  bank  notes  wliicb  circulate  as  a  medium 
of  exchange.  Savings  banks  and  a  few  others  allow 
interest  on  deposits. 

280.  On  opening  an  account  with  a  bank,  the  customer 
is  usually  given  a  pass  book  in  which  the  dates  and 
amounts  of  all  deposits  are  entered  on  the  credit  side.  If 
the  customer  wislies  to  draw  money  from  the  bank,  or  to 
pay  a  debt,  he  fills  out  a  check  similar  to  the  following, 
and  the  dates  and  amounts  of  all  such  checks  are  entered 
on  the  debit  side  of  his  pass  book  : 


^^o 

Kew  Yovh, 

190. __ 

jFourdj  National  Bank 

of  Neto 

godt 

Paij  to  Ruii'Pv 

171.  fSvaaiyyi 

^..or  or  del 

$600.00._. 

S't'V-&  fuvnclv&cl.^ 

no 
luo 

Dollars. 

^&cyvcj.&  €. 

.%x. 

George  E.  Fox  is  the  drawee  or  maker  of  this  check,  and  Ralph 
M.  Brown  is  the  payee. 


281.  As  in  the  case  of  promissory  notes,  checks  may  be 
made  payable  to  payee  or  order,  or  to  payee  or  bearer. 
The  same  rules  of  indorsement  apply.  When  the  deposi- 
tor wishes  to  draw  money  at  the  bank,  the  check  is  made 
payable  to  self. 


188  BANKS  AND  BANKING 

282.    Banks  also  issue  certificates  of  deposit 

Certificate  of  Deposit 


C  ^  =  rf 


g  J;  ox)  « 


^o Ypsilanti,  Mich 190 -. 

ha. ^.deposited  in  the 

jFirst  National  Uaiilt  of  gfpsilanti, 

Dollars, 

payable  to  the  order  of. 

subject  to  the  rxdes  of  this  Banh,  on  tlie  return 
of  this  Certificate,  properly  endorsed. 


Cashier. 


Teller. 


The  money  deposited  on  such  a  certificate  is  not  subject 
to  check  and  can  be  withdrawn  only  upon  the  presentation 
of  the  certificate  properly  indorsed. 


EXERCISE   66 

1.  Write  a  check  for  $35.75  in  each  of  the  forms 
indicated  above,  with  yourself  as  drawer  and  Robert 
Lyons  as  payee.  If  necessar}^,  indorse  the  check  as  when 
presented  for  payment. 

2.  July  22,  1903,  a  man  deposits  fl25  in  a  savings 
bank.  The  bank  pays  3%  on  money  left  on  deposit  3 
mo.,  and  3|%  if  left  6  mo.  or  longer.  Money  must  be 
deposited  the  first  of  the  month  to  draw  interest  for  that 
calendar  month.  If  the  money  is  drawn  out  Nov.  1  and 
interest  paid  for  full  months,  how  much  does  the  man 
receive  ?  if  drawn  out  Dec.  22  ?  if  drawn  out  March  1, 
1904? 


BANKS  AND  BANKING 


189 


3.  A  man  owns  a  certificate  of  deposit  for  f  500  dated 
Aug.  1,  1908.  Feb.  1,  1904,  he  presents  the  certificate 
and  draws  #200.  What  is  the  face  of  the  new  certificate 
issued  ? 

4.  Show  tliat  the  following  statement  of  the  resources 
and  liabilities  of  a  savincrs  bank  will  balance. 


Resources 


Loans  and  discounts 

Bonds,  mortgages  and  securities 

Premiums  paid  on  bonds 

Overdrafts 

Furniture  and  fixtures 

Other  real  estate     . 

Items  in  transit 

Due  from  banks  in  reserve  cities 

Exchanges  for  clearing  house 

U.  S.  and  National  bank  currency 

Gold  coin        .... 

Silver  coin      .... 

Nickels  and  cents  . 


^214,193.14 

20,742.4 

70,470.00 

68,200.00 

2,365.00 

83.28 


$946,542.72 

504,171.35 

1,218.75 

1,471.70 

13,801.50 

85,337.01 

13,409.02 


Liabilities 


Capital  stock  paid  in 
Surplus  fund 
Undivided  profits,  net    . 
Commercial  deposits 
Certificates  of  deposit    . 
Due  to  banks  and  bankers 
Certified  checks 
Cashier's  checks     . 
Savings  deposits    . 
Savings  certificates 


1600,991.33 
94,897.63 

192,674.51 
12,124.52 
10.455.21 

696,425.79 
69,655.31 


1 200,000.00 
36,000.00 
19,781.58 


190  BANKS  AND  BANKING 

283.  If  his  financial  standing  is  high,  a  person  may 
borrow  money  from  a  bank  by  giving  his  individual  note. 
The  bank  may,  however,  ask  for  security.  In  this  case  the 
borrower  must  have  some  responsible  person  indorse  the 
note,  or  he  must  deposit  collateral  security  in  the  form  of 
stocks,  bonds,  etc. 

284.  The  following  is  a  common  form  of  a  bank  note  : 


~  Detroit,  Midi.,  ^W/sl.  6,  19 0¥- 

3^kv&&  viantkoy  after  date,  J---  promise  to  pay 
to  the  order  of_ 

cJ^k&  (k-a^yvyyidyV^ioyt  c/tatiancit  Bank 

3^^v-a  ku.ncLi&cL  and  y^ Dollars 

at  Tlie  Coimnercial  J^atl.  Bank  of  Detroit,  Mich., 
with  interest  at    6   per  cent  per  annum  itntil  due, 
and  seven  per  cent  per  annum  thereafter  until  paid. 
Value  received. 


If  Mr.  Rowe  wishes  to  borrow  $200  he  takes  the  above  note  to 
the  bank  and,  if  necessary,  either  furnishes  an  indorser  or  collateral 
security,  such  as  bonds,  etc.,  which  he  assigns  to  the  bank.  If  the 
bank  authorities  are  satisfied,  he  receives  $200  —  |3  =  8197,  the 
interest  for  3  mo.  being  deducted.  This  interest  is  called  bank 
discount.     Days  of  grace  are  now  rarely  used  by  bankers. 

285.  In  discounting  notes,  banks  count  forward  by  days 
or  months  as  stated  in  the  note  and  usually  reckon  360 
days  in  the  year.  Thus,  a  note  dated  July  22  at  60  days 
will  mature  July  22  +  60  days,  or  Sept.  20,  A  note 
dated  July  22  at  2  mo.  will  mature  Sept.  22. 


BANKS  AND  BANKING  191 


EXERCISE 

67 

Each  of  the  following 

notes  is 

;  discounted  on 

the  date 

of  issue.     Find  the  ( 

late 

of  matui 

•ity 

and  the  discount. 

Date  of  Note 

Time 

Fa(E 

Katk 

1.    Jan.  2,  1905 

60  da. 

11000 

6% 

2.    Aug.  14,  1905 

3  mo. 

$525 

5% 

3.    Aug.  1,  1905 

90  da. 

$387.50 

6% 

4.    April  20,  1905 

30  da. 

$500 

6% 

5.    June  27,  1905 

2  mo. 

$325 

T% 

286.  Business  men  frequently  take  notes  due  at  some 
future  date  from  their  customers,  and  in  case  money  is 
needed  before  the  notes  are  due,  sell  them  to  a  bank. 
(Such  a  note  is  shown  on  page  179.)  The  seller  must 
give  satisfactory  security.  The  bank  pays  the  sum  due 
at  maturity  less  the  discount  from  the  date  of  discount  to 
the  date  of  maturity.  The  sum  paid  by  the  bank  is  called 
the  proceeds.  These  notes  may  or  may  not  bear  interest. 
The  following  examples  will  illustrate  both  cases : 

Ux.  1.  A  note  for  $527.30,  dated  Aug.  31,  1904,  due  in 
90  da.,  without  interest,  was  discounted  at  the  bank 
Oct.  10,  at  6%.     Find  the  proceeds. 

Solution.     Face  of  note  =$527.30 

Discount  for  50  da.  =  4.39 
Proceeds  =  $522.91 

Ux.  2.  A  note  for  $378.50,  dated  Aug.  1,  1904,  due  in 
4  mo.  at  6%,  was  discounted  at  the  bank  Oct.  1,  1904, 
at  6%.     Find  the  proceeds. 

Solution.     Face  of  note  =  $378.50 

Interest  for  4  mo.  =  7.57 
Amount  discounted  =  $386.07 
Discount  for  2  mo.  =  3.86 
Proceeds  =  $382.21 


192  BANKS  AND   BANKING 

EXERCISE   68 

Find  the  discount  and  proceeds  of  each  of  the  following 
non-interest  bearing  notes : 

Time 

30  da. 

2  mo. 
90  da. 
60  da. 

3  mo. 

Find  the  discount  and  proceeds  of  each  of  tlie  following 
interest-bearing  notes : 

„  T^  m  Rate  of         Date  of         Rate  of 

Face  Date  Time         t  t^  t^ 

Interest        Discount       Discount 


Face 

Date 

1. 

$500 

July  1 

2. 

•1225.75 

April  10 

3. 

$253.30 

Dec.  14 

4. 

1150.40 

Aug.  12 

5. 

11250 

Nov.  1 

Date  of 
Discount 

Rate  of 
Discount 

July  10 

7% 

May  1 

6% 

Jan.  2 

6% 

Sept.  5 

5% 

Dec.  1 

6% 

6. 

11500 

Aug.  10 

90  da. 

6% 

Sept.  1 

7% 

7. 

$97.30 

Oct.  2 

60  da. 

7% 

Nov.  1 

6% 

8. 

1152.20 

Sept.  4 

4  mo. 

5% 

Oct.  20 

6% 

9. 

$750.50 

Jan.  4 

30  da. 

4% 

Feb.  1 

7% 

10. 

$431.40 

June  17 

2  mo. 

6% 

July  10 

6% 

11.  A  merchant's  bank  account  is  overdrawn  $2150.75, 
and  he  presents  to  the  bank  the  following  notes,  which  are 
discounted  Dec.  5  at  6%  and  placed  to  his  credit.  What 
is  his  balance  ?    . 

Face  Date  Time  f'^™  ^^ 

Interest 

$500  Nov.  12         60  da.         5% 

$1250.25         Sept.  30         90  da.         4% 
$727.40  Oct.    25         3  mo.  no  interest 

12.  For  how  much  must  I  give  my  note  at  the  bank, 
discounted  at  6%  and  due  in  60  da.,  to  realize  $1500? 

Sur/fjestum.  Find  the  proceeds  of  <f  1  discounted  at  6%  for  00  da. 
and  divide  |1500  by  the  result. 


EXCHANGE 


287.  The  subject  of  exchange  treats  of  methods  of  can- 
celing indebtedness  between  distant  places  without  the 
actual  transfer  of  money.  This  may  be  accomplished  in 
any  one  of  the  following  ways : 

(1)  By  Postal  Money  Order.  Money  orders  may  be  sent 
for  any  amount  from  1  ct.  to  tjt5lOO. 

Identification  by  payee  is  necessary  unless  the  sender 
waives  it.  In  case  of  payment  to  the  wrong  person,  the 
government  cannot  be  sued.  At  present  the  rates  charged 
are  as  follows : 

For  orders  for  sums  not  exceeding  -f  2.50 
If  over  $2.50  and  not  exceeding  ^5.00 
If  over  $5.00  and  not  exceeding  110.00 
If  over  $10.00  and  not  exceeding  $20.00 
If  over  $20.00  and  not  exceeding  $30.00 
If  over  $30.00  and  not  exceeding  $40.00 
If  over  $40.00  and  not  exceeding  $50.00 
If  over  $50.00  and  not  exceeding  $60.00 
If  over  $60.00  and  not  exceeding  $75.00 
If  over  $75.00  and  not  exceeding  $100.00 

(2)  By  Express  Money  Order.  The  rates  are  the  same  as 
for  postal  orders.  The  company  is  responsible  for  the 
payment  to  wrong  persons. 

(3)  By  Telegraphic  Money  Order.  Telegraph  companies 
charge  1%  of  the  amount  sent,  with  a  minimum  charge 
of  25  ct.,  and  double  the  charge  for  a  15-word  message 
between  the  sending  office  and  the  office  of  the  District 
Superintendent,  through  whose  office  the  order  is  sent. 

ltman's  adv.  ar.  — 13  193 


3  cents. 

5  cents. 

8  cents. 

.    10  cents. 

.    12  cents. 

.     15  cents. 

.     18  cents. 

.    20  cents. 

.    25  cents. 

.    30  cents. 

194  EXCHANGE 

(4)  By  Check.  A  personal  check  on  a  home  bank  where 
the  sender  has  money  deposited  can  be  sent.  This  check, 
when  properly  indorsed  by  the  payee  and  presented  at 
his  bank,  will  probably  be  cashed  without  charge.  The 
bank  may  charge  a  small  fee,  called  exchange,  for  collecting. 

(5)  By  Bank  Draft.  The  ordinary  form  of  bank  draft 
is  as  follows : 


JVo.  ^^,786 

Central  Sabtngs;  Bank 


^  Detroit,  Mich.,  oie^jol.  6,  1905 

Pay  to  the  order  of  ^ciyyvt^  /if.  /CclM& 

'Qn^&  k^mcLvtcL  Uivviy^-jCv-&  a^ncl  j^ Dollars. 

To  The  Fourth  JVational  Bauh  1 
of  the  City  of  JVew  Yorh.        J 


The  draft  differs  from  the  check  in  that  it  is  drawn 
by  one  bank  on  another,  while  tlie  check  is  drawn  by  an 
individaal  on  a  bank  where  he  has  money  deposited. 

288.  Most  banks  keep  money  on  deposit  in  some  bank 
called  a  correspondence  bank,  in  a  large  commercial  center 
like  New  York  or  Chicago. 

If  a  customer  of  a  local  bank  in  the  West  wishes  to  pay  a  debt  in 
the  East,  he  buys  a  draft,  signed  by  the  cashier  of  the  local  bank,  and 
drawn  against  the  correspondence  bank  in  New  York  City.  This 
draft  will  pass  as  cash  at  any  bank.  The  above  draft  is  drawn  by  the 
Central  Savings  Bank  of  Detroit,  and  the  correspondence  bank  is  the 
Fourth  National  Bank  of  the  City  of  New  York. 

289.  The  following  is  another  form  of  a  draft,  called 
the  commercial  draft. 


EXCHANGE  195 


$/ooom 

J^eiv  York,  CUcf.  /o,  1903. 

At  sight  pay  to  the 

order  of  first  i^fational  Bank 

€.n&  tkatc^ancL  cLivd.. 

7»0 

riw  Dollars. 

l\ih(e  received,  and  cJuirge  the  same  to  the  aeeoiuit  of 

To  /{-ci^v^te.^  V  (go.. 

(X^}v&vC(^AA^'^^  fSook^  (^a. 

^elvait,  mUL 

Hawkes  &  Co.  owe  the  American  Book  Co.  !$1000  past  due.  Tlie 
American  Book  Co.  draws  the  above  sight  draft  payable  to  the  order 
of  the  First  National  Bank  of  New  York,  and  deposits  it  for  collec- 
tion. The  First  National  Bank  sends  it  to  a  Detroit  bank  to  collect. 
The  Detroit  bank  demands  payment  of  Hawkes  &  Co.  If  payment  is 
made,  the  money  is  remitted  to  New  York.  If  payment  is  refused,  the 
draft  is  returned  to  the  New  York  bank,  and  the  American  Book  Co. 
is  notified.     Some  other  means  of  collecting  must  then  be  employed. 

290.  If  the  account  against  Hawkes  &  Co.  were  not  due 
for  60  days,  the  draft  woukl  read  as  follows : 


$ /000m  Xeiv  York,  (Zucf.    fS,  190^. 

At  sixty  days  sight  pay  to  the  order  of  JFirst  U^Tational 

Bank  iln&  tko-^oa.ci')^cL  a^n.cL Too  Dollars. 

Value  received,  and  charge  the  same  to  the  account  of 

To  f{-oL^v~ke.^  V  ta.  CLy>te.vUa.^v  Baa/o  (Eo-. 


196  EXCHANGE 

This  draft  is  called  a  time  draft  and  would  be  taken  to  Hawkes  &  Co. 
as  before,  who,  if  they  intended  to  pay  it,  would  write  across  the  face 
in  red  ink  :  Accepted  Aug.  21,  1903. 

Hawkes  Sf  Co. 

After  writing  these  w^ords  across  the  draft,  Hawkes  &  Co.  have 
agreed  to  pay  $1000,  and  the  draft  becomes  the  same  as  a  promissory 
note. 

291.  Fluctuations  of  Exchange.  If  the  banks  of  San 
Francisco  have  sokl  drafts  for  a  larger  sum  on  the  New 
York  banks  than  they  have  on  deposit  in  New  York,  it 
will  be  necessary  to  send  money  enough  to  New  York  to 
balance  the  account.  The  money  is  usually  sent  by  ex- 
press at  some  expense,  which  must  be  borne  by  the  pur- 
chaser of  the  drafts.  In  this  case  a  draft  on  New  York 
would  cost  slightly  in  excess  of  its  face.  This  excess  is 
called  a  premium.  A  draft  sold  at  less  than  its  face  is  said 
to  be  sold  at  a  discount. 

Premiums  and  discounts  are  usually  quoted  as  a  per  cent  of  the 
face  of  the  draft.  Thus,  a  quotation  of  ^^  %  premium  means  that  a 
draft  for  .f  100  may  be  purchased  for  -1100.10.  Sometimes  the  quota- 
tion is  a  certain  amount  per  $1000.  Thus,  if  New  York  exchange  is 
quoted  at  $1.50  premium  at  New  Orleans,  a  draft  for  $1000  will  cost 
$1001.50. 

The  above  quotations  refer  to  sight  drafts.  Time  drafts  are  dis- 
counted by  banks  in  the  same  manner  as  promissory  notes. 

New  York  City  is  the  greatest  financial  center  of  the  United  States, 
and  so  much  business  is  transacted  through  the  New  York  banks  that 
New  York  exchange  is  generally  at  a  premium.  Consequently  banks 
are  always  willing  to  cash  such  checks  at  par  value.  People  in  New 
York  usually  pay  their  indebtedness  outside  of  the  city  by  checks  or 
drafts  on  New  York  banks,  which  find  a  ready  sale  at  any  bank. 

292.  The  Clearing  House  is  an  institution  organized  by 
the  banks  of  every  large  city  to  facilitate  settlement  of 
claims  against  one  another. 


EXCHANGE  107 

Clerks  from  each  bank  bring  daily  to  tlie  clearing  house  the 
checks,  etc.,  due  them  from  all  other  associated  banks,  each  bank  being 
represented  by  a  separate  package.  Balances  are  struck  between  the 
credits  and  debits  of  each  bank  against  all  others,  and  the  manager 
certifies  the  amount  which  each  bank  owes  to  the  associated  V)anks  or 
is  entitled  to  receive  from  them.  The  banks  whose  debits  exceed 
their  credits  pay  in  the  balance  to  the  clearing  honse,  which  issues 
clearing  house  certificates  to  the  banks  whose  credits  exceed  their 
debits.  In  the  New  York  Clearing  House,  which  is  the  largest  in 
America,  nearly  sixty  billion  dollars  of  clearings  were  made  in  1904, 
with  only  three  billions  of  dollars  of  balances  paid  in  money. 

EXERCISE   69 

1.  A  quotation  of  #  2.50  preniiuni  is  equivalent  to  what 
per  cent  ? 

2.  What  is  the  cost  in  Kansas  City  of  a  draft  on  New 
York  for  $  67.50  at  \  f  premium  ? 

3.  What  is  the  cost  in  Galveston  of  a  draft  on  New 
York  for  ^4380.50  at  12.50  premium  ? 

4.  In  Ex.  2  and  3  which  city  is  owing  the  other  money? 

5.  Find  the  cost  of  a  draft  for  f  500  payable  in  60  days 
after  sight,  exchange  being  \  Jo  premium,  interest  6  Jo. 

6.  Find  the  cost  of  sending  #67.50  by  telegraphic  money 
order  if  a  10-word  message  costs  40  ct.  and  each  word  in 
excess  of  10  costs  2  ct. 

*  7.    How  much  would  it  cost  to  send  the  same  amount 
by  postal  money  order  ?  by  express  money  order  ? 

8.  A  draft  on  New  York  for  810000  costs  89980  in 
Chicago.  Is  exchange  at  a  premium  or  a  discount?  What 
is  the  rate  of  exchange  ?  The  balance  of  trade  is  in  favor 
of  which  city  ? 

9.  A  merchant  has  a  60-day  note  for  81200  discounted 
at  the  bank  at  6  Jo  and  purchases  a  draft  with  the  proceeds, 


198  EXCHANGE 

exchange  -fl.OO.      He    sends   the   draft  to  a  creditor  to 
apply  on  account.     How  much  is  phiced  to  his  credit  ? 

10.  A  Chicago  banker  discounts  a  draft  for  -^2500,  pay- 
able at  St.  Louis  60  days  after  siglito  What  are  the  pro- 
ceeds, exchange  at  |^  %  discount,  interest  6  %  ? 

FOREIGN  EXCHANGE 

293.  Foreign  exchange  is  the  same  in  all  essential  fea- 
tures as  domestic  exchange,  the  difference  being  that 
exchange  takes  place  between  cities  in  different  countries. 

294.  A  draft  on  a  foreign  country,  usually  called  a  bill 
of  exchange,  is  payable  in  the  currency  of  the  country  on 
which  it  is  drawn. 

295.  Foreign  bills  of  exchange  are  generally  written  in 
duplicate,  called  a  set  of  exchanges,  and  are  of  the  follow- 
ing form  : 


JYew  York,  CUicf,  /^,  1903. 
Exchange  for  £100. 

^&7^  clciyQ.  after  siglit  of  this  first  of  exchange 
(second  of  same  date  and  tenor  unpaid)  pay  to  the 

order   of €.    /if.    Tfl&n^d 

cy}i,&    kit  ncli  &ct    jilaa^.')^cL^    at&i  tvytcf, 

and  charge  the  same,  ivithout  further  advice,  to 

To   fSavi^cf   JSvcytkeAA.,  ^t^cyuje.  €.   S'cyx.. 

jCo-nclcy/b. 
Jfo.  f 736^6. 


FOREIGN   EXCHANGE  199 

The  duplicate  sLil)stitiites  "second  of  exchange"  for  *' first  of  ex- 
change "  and  "  first  of  same  date"  for  "  second  of  same  date,"  in  the 
original.     Eillier  one  being  paid,  the  other  becomes  void. 

296.  1  he  par  of  exchange  between  two  countries  is  the 
value  of  the  monetary  unit  of  one  expressed  in  Unit  of  tlie 
other.  Thus,  the  gokl  in  the  English  pound  is  worth 
if) 4. 8665.  Exciiange  on  Paris  and  other  countries  using 
the  French  monetary  system  is  usually  quoted  at  so  many 
francs  to  the  dollar,  sometimes  at  so  many  cents  to  the 
franc.  The  par  of  exchange  is  about  5.18^  francs  to  the 
dollar,  or  19.3  cents  to  the  franc.  Exchange  on  Germany 
is  quoted  at  so  many  cents  on  4  marks.  The  par  of  ex- 
change is  95.2 ;   quoted  in  cents  per  mark  it  is  23.8. 

EXERCISE   70 

1.  What  is  the  cost  of  a  draft  on  London  for  £  150, 
exchange  §4.925? 

2.  What  is  the  cost  of  a  draft  on  Paris  for  1200  francs, 
exchange  5.20? 

3.  In  either  Ex.  1  or  2  is  the  balance  of  trade  in  favor 
of  the  United  States? 

4.  A  tourist  purchased  a  letter  of  credit  and  drew  £  80 
at  London,  1500  francs  at  Paris,  750  marks  at  Berlin. 
How  much  did  the  letter  cost  him  if  exchange  is  |  % 
premium  at  London,  |  %  premium  at  Paris,  and  ^  %  pre- 
mium at  Berlin? 

5.  What  is  the  cost  of  a  draft  on  Leipsic  for  525  marks, 
exchange  96?    exchange  24? 

6.  What  is  the  cost  of  a  draft  on  London  for  £  75, 
exchange  rif4.857? 


STOCKS   AND   BONDS       . 

297.  A  corporation  or  stock  company  is  an  association  of 
individuals  under  the  laws  of  a  state  for  the  purpose  of 
transacting  business  as  one  person.  Large-scale  produc- 
tion is  now  usually  conducted  by  corporations. 

A  corporation  is  managed  by  officers  elected  by  aboard  of  directors 
who  are  chosen  by  the  stockholders,  each  stockholder  having  as  many 
votes  as  he  owns  shares  of  stock.  The  capital  stock  is  divided  into  a 
certain  number  of  shares,  the  par  value  of  which  is  determined  by  the 
number  of  shares  into  which  the  stock  is  divided.  Thus,  a  capital 
stock  of  -150000  may  be  divided  into  500  shares  of  $100  each,  or  2000 
shares  of  -125  each,  etc.  Stockholders  may  own  any  number  of  shares 
and  participate  in  the  profits  according  to  the  number  of  shares  they 
own. 

298.  If  a  company  is  prosperous  and  makes  more  than 
its  expenses,  a  dividend  is  paid  to  the  stockholders.  The 
dividend  is  usually  a  certain  per  cent  of  the  par  value  of 
the  stock,  or  sometimes  so  many  dollars  per  share.  If  the 
rate  of  dividend  is  higher  than  the  current  rate  of  interest, 
there  usually  will  be  a  demand  for  the  stock  and  it  will 
sell  at  a  premium.  If  the  rate  of  dividend  is  lower,  the 
demand  will  be  sliofht  and  the  stock  will  sell  at  a  discount. 


& 


299.  Companies  frequently  issue  two  kinds  of  stock,  preferred  and 
commoji.  The  holders  of  preferred  stock  are  entitled  to  first  share  in 
the  net  earnings  of  the  corporation  up  to  a  certain  amount,  usually 
from  5%  to  7%  of  the  par  value.  The  holders  of  common  stock  are 
entitled  to  a  share,  or  all  of  what  is  left  after  tjie  dividend  on  the 
preferred  stock  is  paid. 

200 


STOCKS  AND  BOXDS  201 

Stock  is  sometimes  issued  to  the  stockholders  of  a  corporation 
without  a  corresponding  increase  in  the  vahie  of  the  property.  Such 
stock  is  called  watered  stock.  A  corporation  may  be  prohibited  by 
its  charter,  or  by  law,  from  paying  dividends  in  excess  of  a  certain 
amount.  ThuSj  if  a  corporation  with  a  capital  stock  of  «|  100000 
makes  $16000  and  wishes  to  pay  this  amount  in  dividends,  but  is 
prohibited  from  paying  more  than  8  %,  watered  stock,  equal  in  amount 
to  the  capitalization  of  the  corporation,  may  be  issued  to  the  stock- 
holders and  an  8%  dividend  (=  |  IGOOO)  may  be  declared  upon  this 
new  basis. 

300.  Since  it  is  difficult  for  individuals  to  buy  and  sell 
stock  personally,  the  business  is  usually  transacted  through 
a  stock  broker,  who  charges  a  small  per  cent,  called  broker- 
age (usually  J%  ),  of  the  par  value  of  the  stock  bought  or 
sold.  The  stock  broker  generally  belongs  to  an  organi- 
zation called  a  stock  exchange.  The  New  York  Stock 
Exchange  is  the  most  important  in  America. 

301.  Generally  each  stockholder  is  responsible  only  to  the  extent 
of  the  par  value  of  the  stock  he  owns.  In  the  case  of  national  banks, 
however,  a  stockholder  is  liable  to  the  amount  of  the  par  value  of  his 
stock  in  addition  to  the  amount  paid  for  the  purchase  of  the  stock. 

302.  Investors  often  buy  stocks  and  hold  them  for  the  dividend 
they  yield.  Speculators  buy  them  to  sell  at  a  profit.  Speculators 
usually  buy  on  a  margin,  that  is,  they  pay  only  a  part  of  the  purchase 
price  and  borrow  the  rest  by  depositing  the  certificate  of  stock  as 
cojlateral.  A  man  who  buys  stock  on  a  20-point  margin  pays  down 
20%  of  the  par  value  and  borrows  the  rest.  A  "bull"  is  a  buyer  of 
stocks  which  he  hopes  to  sell  at  a  profit.  He  acts  on  the  belief  that 
prices  will  go  up.  A  "bear"  is  a  seller  of  stocks  which  he  does  not 
possess,  but  borrows  on  the  belief  that  prices  will  go  down.  Tlius,  if 
a  stock  is  quoted  at  50,  a  "  bear,"  thinking  it  will  go  down  to  45,  may 
sell  at  50,  and  deliver  borrowed  stock  to  the  purchaser.  If  the  stock 
goes  down  to  45,  he  will  purchase  it  and  return  it  to  the  owner,  thus 
realizing  a  profit  of  5  points.  This  is  called  selling  stocks  "  short." 
Bears  are  said  to  be  "  short "  of  stock  and  bulls  "  long." 


202 


STOCKS  AND  BOXDS 


303.  When  for  any  reason  a  stock  company  finds  the 
amount  of  money  paid  in  by  stockholders  insufficient,  it 
generally  borrows  money  and  issues  bonds,  secured  by  a 
mortgage  on  the  property  of  the  company.  These  bonds 
are  written  agreements  to  pay  a  certain  sum  of  money 
within  a  stated  time  and  at  a  fixed  rate  of  interest.  Bonds 
have  a  prior  claim  over  any  kind  of  stock. 

304.  National  governments,  states,  counties,  and  cities 
often  issue  bonds,  but  without  mortgages,  the  credit  of 
such  organization  being  considered  good. 

Registered  bonds  are  issued  in  the  name  of  the  owner,  and  are 
made  payable  to  him  or  his  assignee.  Interest,  when  due,  is  sent 
direct  to  the  owner. 

Coupon  bonds  are  payable  to  bearer,  and  have  small  interest  coupons 
attached,  which  are  cut  off  when  due,  and  the  amount  of  interest  is 
collected  either  personally,  or  through  a  bank.  There  is  a  coupon  for 
each  interest  period. 

Bonds  are  usually  named  from  the  rate  of  interest  they  bear,  or 
from  the  date  at  which  they  are  payable.  Thus,  Union  Pacific  4's 
means  Union  Pacific  bonds  bearing  4%  interest.  U.S.  4's  reg.  1907 
means  United  States  registered  bonds  bearing  4%  interest  and  due  in 
1907.  Western  Union  7's  coup.  1900  means  Western  Union  coupon 
bonds  bearing  7%  interest  and  due  in  1900. 

305.  The  following  quotations  show  the  prices  paid  for 
stocks  and  bonds  on  a  certain  day.  The  daily  newspaper 
will  furnish  the  best  source  for  quotations.. 


Stocks 

Bonds 

Amalgamated  Copper 

. 

72| 

U.S.  New  4's  reg.    .     . 

.  135i 

A.  T.  and  S.  F.  .     .     . 

88| 

U.S.  New  4's  coup. 

.  136i 

A.  T.  and  S.  F.  preferi 

■ed     . 

98| 

U.S.  3's  reg 

.  107i 

Canadian  Pacific     .     . 

. 

131 

U.S.  3's  coup.      .     .     . 

.  108 

National  Biscuit  Co.   . 

. 

98| 

Atchison  4's  .     .     .     . 

•  10-21 

National    Biscuit    Co. 

pre- 

N.Y.  Central  3j's    .     . 

.  103| 

ferred 

. 

105  J 

C.  B.  and  Q.  4's      .     . 

.     93^- 

N.Y.  Central     .     .     . 

. 

143 

C.  R.  I.  and  P.  4's  .     . 

.  105 

STOCKS  AND  BONDS 


203 


Stocks 

Railway  Steel  Spring.     .     . 
Railway  Steel    Spring   pre- 
ferred      

U.S.  Steel  ........ 

U.S.  Steel  preferred     .     .     . 

\Vabash 

Wabash  prefei-red  .     .     .     . 
AVesteru  Union  preferred    . 


33] 

872^ 
37i 
37^ 


Bonds 

Southern  Ry.  5's     .  . 
Detroit  Gas  Co,  .5's 
Chicago  and  Alton  3]'s 

Hocking  Valley  4^'s  . 

B.  and  O.  4's  .     .     .  . 

U.S.  Steel  5's      .     .  . 


116 
lOOi 
70J 
100^ 

mi 


306.  Quotations  are  usually  made  at  a  certain  per  cent 
of  the  par  value  of  the  stock  or  bond.  Thus,  the  quota- 
tion of  72|  for  Amalgamated  Copper  means  72|%  of  the 
par  value  of  one  share.     The  purchaser  must  pay  his  broker 

72^ 
T2| 


+ 1^  =  72|,  and  the  seller  will  receive  from  his  broker 


95 

-8- 


307.  In  the  following  examples  the  par  value  of  a  share 
will  be  taken  as  -$100  unless  otherwise  stated.  Brokerage 
at  1^%  must  be  taken  into  account  in  each  case  where  not 
otherwise  stated. 

Ux.  1.  A  person  buys  100  shares  of  A.  T.  and  S.  F.  as 
quoted  above,  and  sells  6  mo.  later  for  85|^,  having  received 
a  dividend  of  2%.  Does  he  gain  or  lose,  and  how  much, 
money  being  w^orth  6%  per  annum? 


Solution.     Each  share  costs 
and  is  sold  for 

.-.  the  gain  on  each  share  is 
.-.  the  gain  on  100  shares  is 
The  dividend  received  = 
.-.  the  total  gain  is 
The  amount  invested  is 


83f  +  i  =  83J, 
851  _  1  =.  85. 
85-83^  =  1^,  or  $1.50. 
100  X  11.50  =  ^150. 
2%  of  $10000  =  1200. 
$150 +  $200  =  $350. 
100  X  $83^  =  $8350. 


The  interest  for  6  mo.  is  h  of  6%  of  $8350  =  $250.50. 

.-.  $350  -  $250.50  =  $99.50  =  the  net  gain. 


204  STOCKS  AND  BONDS 

Ex.  2.  A  man  sells  short  100  shares  of  Canadian  Pacific 
at  131  and  three  days  later  *'  covers  "  (that  is,  buys  the 
stock)  at  128|.     What  is  his  net  profit  ? 

Solution.     Each  share  sold  yields  131    ~\  =  $130| 
Each  share  is  bought  for  I28f  +  i  =     128| 

Therefore  the  gain  on  each  share  is  f  2 

Therefore  the  net  gain  on  100  shares  is  100  x  ^2  =  |200. 

EXERCISE   71 

1.  The  capital  stock  of  a  company  is  f  1000000,  ^  of 
which  is  preferred,  entitled  to  a  7  %  dividend,  and  the  rest 
common.  If  147500  is  distributed  in  dividends,  what  rate 
of  dividend  is  paid  on  the  common  stock  ? 

2.  A  person  buys  302  shares  of  stock,  par  value  810,  for 
17  a  share,  paying  5  ct.  a  share  brokerage.  6  mo.  later, 
after  having  received  a  5%  dividend,  he  sells  for  89.75  a 
share.    How  much  does  he  make,  money  being  worth  6%  ? 

3.  If,  in  Ex.  1, 1  77500  is  distributed  in  dividends,  which 
is  the  better  stock  to  own,  common  or  preferred  ? 

4.  Which  is  the  better  property  to  own,  8  1000  stock  in 
a  company  at  6  %,  or  one  of  its  8 1000  bonds  at  4  %  ? 

5.  Which  is  the  safer  against  loss  by  theft,  a  coupon 
bond,  or  a  registered  bond  ?  Which  is  the  more  readily 
transferred  ? 

6.  Why  are  U.S.  4's  registered  quoted  at  135],  while 
U.S.  4's  coupon  are  quoted  at  136^^? 

7.  How  much  will  50  shares  of  Amalgamated  Copper 
cost  ? 

8.  How  much  will  75  Atchinson  4's  cost  ? 

9.  How  much  will  100  shares  of  New  York  Central 
cost? 


STOCKS  AND  BONDS  205 

10.  How  miicli  will  100  New  York  Central  3.]  %  bonds 
cost  ? 

11.  Which  should  you  prefer  to  own,  the  100  shares  of 
stocks  or  the  100  bonds  mentioned  in  Ex.  9  and  10  ? 

12.  What  is  the  greatest  number  of  Canadian  Pacitic 
shares  that  can  be  purchased  for  ^  5000  ? 

13.  Which  is  the  better  investment,  a  4%  mortgage  or 
Southern  5's  as  quoted  ? 

14.  Which  is  the  better  investment,  B  and  O.  4's  or 
U.S.  5's  as  quoted? 

15.  What  sum  must  be  invested  in  Atchison  4's  at  102| 
to  secure  an  annual  income  of  84120  ? 

16.  What  rate  of  income  will  U.S.  3's  registered  yield  ? 

17.  If  I  pay  $3762.50  for  U.S.  Steel  preferred,  how 
many  shares  do  I  buy  ? 

18.  How  much  must  I  pay  for  B.  and  O.  4's  to  yield  an 
income  of  5%  on  my  investment?    of  6%  ? 

19.  What  income  will  a  man  receive  from  an  invest- 
ment of  $21625  in  U.S.  3's  coup.  ? 

20.  What  dividend  can  a  company  declare  on  a  capital 
stock  of  $50000  whose  net  earnings  are  $7500? 

•  21.  A  certain  bank  pays  a  semiannual  dividend  of  3^% 
on  its  stock  ;   what  is  the  annual  dividend  on  25  shares  ? 

22.  How  much  must  I  pay  for  5%  bonds  that  the 
investment  may  yield  6%  income?  for  4%  bonds?  for 
3%  bonds? 

23.  A  man  owms  100  shares  of  Amalgamated  Copper 
stock.  If  the  company  declares  a  dividend  of  5%  payable 
in  stock,  how  much  stock  will  he  then  own  ? 


206  STOCKS  AND  BONDS 

24.  My  broker,  after  selling  for  me  200  shares  of 
Wabasli  preferred,  remitted  to  me  i^9987.50.  At  what 
price  did  he  sell  the  stock  ? 

25.  How  much  must  be  invested  in  U.S.  3"s  coup,  to 
bring  an  annual  income  of  $  500  ? 

26.  A  bank  with  a  capital  stock  of  1 150000,  declares  a 
semiannual  dividend  of  3%.  What  is  the  amount  of  the 
dividend,  and  how  much  will  a  person  receive  who  owns 
25  shares  ? 

27.  A  gas  company  declares  a  6%  dividend  and  dis- 
tributes $120000  among  its  stockholders.  What  is  its 
capital  stock  ? 

28.  A  cement  company  divides  ^^  80000  among  its  stock- 
holders. What  is  the  rate  of  dividend,  the  capital  stock 
being  $  1000000  ?  How  much  is  paid  to  a  person  who  owns 
902  shares  of  ^  10  each  ? 

29.  A  broker  bought  for  a  customer  500  shares  of  cop- 
per stock,  par  value  125,  at  a  total  cost  of  818015.63. 
Find  the  market  quotation  and  brokerage. 

30.  A  man  bought  200  shares  of  New  York  Central  at 
143.  The  market  price  declined  till  it  reached  139  and 
then  rallied  to  141^.  Believing  that  another  decline  was 
coming,  he  sold  500  shares  (300  of  them  short)  at  1411. 
The  price  continued  to  rally,  however,  and  he  covered  by 
buying  300  shares  at  1421.  What  was  the  net  loss  on 
the  whole  transaction,  making  no  allowance  for  interest, 
but  allowing  |%  brokerage  for  each  sale  and  purchase? 


INSURANCE 

308.  There  are  two  general  classes  of  insurance  :  in- 
surance on  the  person  in  the  form  of  life,  endowment, 
accident,  and  health  insurance,  and  insurance  on  property 
in  the  form  of  fire,  marine,  live  stock,  tornado,  plate  glass, 
boiler  insurance,  insurance  against  bad  debts,  etc. 

PROPERTY    INSURANCE 

309.  Tlie  principal  kinds  of  property  insurance  are  fire 
insurance,  or  insurance  against  loss  by  fire ;  marine  insur- 
ance, or  insurance  against  loss  of  vessels  at  sea,  or  property 
on  board  of  vessels  at  sea  ;  tornado  insurance,  or  insurance 
against  loss  by  storms,  etc. 

310.  The  written  agreement  between  the  company  and 
the  person  insured  is  called  the  policy,  and  the  sum  to  be 
paid  by  the  company  in  case  of  loss,  the  face  of  the  policy. 
The  person  insured  is  called  the  insured,  and  the  amount 
paid  by  the  insured  to  the  company  for  insurance,  the 
premium. 

311.  The  premium  is  usually  computed  as  a  certain  per 
cent  of  the  face  of  the  policy,  or  as  a  certain  sum  on  each 
8100  of  insurance.  In  either  case  it  is  called  the  rate  of 
insurance. 

Ex.  A  house  valued  at  85000  is  insured  for  |  of  its 
value  at  1.1%  per  annum.     What  is  the  annual  premium  ? 

207 


208  INSURANCE 

How  much  would  the  owner  lose  if  the  house  were  burned 
after  seven  premiums  had  been  paid  ?  How  much  would 
the  company  lose  ? 

Solution.     Valuation  of  house  is  f  of  ^5000  =  $4000. 
Premium  =  1.1  %  of  $4000  =  $44. 

Loss  of  owner       =  $5000  -  $4000  +  7  x  $44  =  $l308o 
Loss  of  company  =  $4000  -  7  x  $44  =  $3692. 

The  above  rate  of  insurance  might  have  been  stated  as 
11.10  on  each  |100  insured. 

EXERCISE  72 

1.  A  house  valued  at  16000  is  insured  for  |  of  its  value 
at  1%  per  annum.  What  is  the  annual  premium  ?  How 
much  does  the  owner  lose  if  the  house  is  burned  after  10 
premiums  have  been  paid  ?  How  much  does  the  company 
lose? 

2.  How  much  would  the  owner  lose  in  case  the  house 
were  damaged  by  fire  to  the  extent  of  il500  after  3  pre- 
miums had  been  paid  ? 

3.  How  much  would  the  owner  lose  if  the  house  were 
damaged  by  fire  to  the  extent  of  ^^350  after  9  premiums 
had  been  paid  ? 

4.  A  residence  valued  at  $3500  is  insured  for  |  of  its 
full  value  at  I  %  per  annum.  The  company  will  insure  the 
house  for  3  yr.  on  the  payment  of  2^  times  the  annual 
premium  in  advance.  How  much  will  it  cost  to  insure 
the  house  for  3  yr.  ?  They  will  insure  for  5  yr.  on  the 
payment  of  4  times  the  annual  premium  in  advance. 
How  much  will  it  cost  to  insure  the  house  for  5  3^-.  ? 


PBOPEllTY  INSURANCE  209 

5.  How  much  will  it  cost  to  insure  a  manufacturing 
plant  valued  at  !i^ 65000  at  |%  and  the  machinery  valued 
at  . 130000  at  ^9^%? 

6.  The  insurance  in  Example  5  is  placed  in  f(nir  compa- 
nies, as  follows  :  building,  125000,  i  20000, 115000, 15000; 
machinery,  112000,  ^8000,  *(.;000,  14000.  What  is  the 
annual  premium  paid  each  company  ? 

7.  A  farmer  takes  the  following  insurance  on  his  prop- 
erty :  house  valued  at  $2500  at  !{  % ;  barn  valued  at  %  1800 
at  1 1  % ;  live  stock  valued  at  ^2600  at  -|% ;  grain  valued  at 
$1800  at  1%;  he  also  takes  tornado  insurance  for  13000 
and  paj^s  40  ct.  per  -SlOO  for  5  yr.  He  pays  4  times  the 
annual  premium  for  fire  insurance  for  5  yr.  and  3  times 
for  live  stock  insurance.  What  is  his  total  premium  for 
5  yr.  ? 

8.  A  dealer  in  Buffalo  ordered  his  Chicago  agent  to 
buy  4000  bu.  of  wheat  at  72  ct.,  2500  bu.  of  oats  at  26  ct., 
7200  bu.  of  corn  at  37 J  ct.,  paying  2%  commis*sion  for 
buying.  The  grain  was  shipped  by  boat,  and  a  policy  at 
1J%  taken  to  cover  the  cost  of  grain  and  all  charges. 
Wliat  Avas  the  amount  of  the  policy  and  what  was  the 
premium  ? 

9.  In  a  town  where  the  regular  police  force  consists  of 
20*  or  more  patrolmen  a  company  will  insure  a  bank  against 
burglary  for  1  yr.  for  50  ct.  per  $100  up  to  $3000,  and 
25  ct.  per  $100  above  that  amount.  How  much  Avill  it 
cost  to  insure  a  bank  for  $50000  against  burglary  in  such 
a  town  ? 

10.  The  annual  premium  for  insuring  a  plate  glass  win- 
dow 6  ft.  by  10  ft.  is  $3.30.  How  much  will  it  cost  a 
merchant  to  insure  two  such  windows  for  5  yr.  ? 

LYMAX'S  ADV.   AR.  14 


210  INSURANCE 

LIFE   AND  ACCIDENT   INSURANCE 

312.  Life  insurance  is  an  agreement  to  pay  to  the  heirs 
of  a  person  a  specified  sum  upon  his  death. 

313.  Endowment  insurance  is  an  agreement  to  pay  a 
specified  sum  to  the  person  insured  if  living  at  the  end  of 
a  definite  period  of  years,  or  to  his  heirs  in  case  of  death 
within  that  period. 

314.  Accident  insurance  is  an  agreement  to  pay  the 
person  insured  a  weekly  indemnity  for  loss  of  time  while 
incapacitated  from  accident,  or  a  fixed  amount  in  case  of 
permanent  injury,  such  as  the  loss  of  both  hands,  both  feet, 
the  entire  sight  of  the  eyes,  etc.,  or  a  fixed  amount  to  his 
heirs  in  case  of  death  by  accident. 

315.  Health  insurance  is  an  agreement  to  pay  a  weekly 
sum  in  case  of  sickness  from  specified  diseases.  In  addi- 
tion to  the  weekly  indemnity,  health  insurance  sometimes 
guarantees  the  payment  of  all  doctor's  fees  and  special 
amounts  to  cover  cost  of  surgical  operations. 

316.  The  following  tables  show  the  annual  rates  per 
$1000  charged  by  one  of  the  leading  life  insurance  com- 
panies doing  business  in  the  United  States.  These  rates 
are  for  life  and  endowment  policies.  Insurance  companies 
also  issue  rates  payable  semiannually  or  quarterly.  Such 
rates  are  slightly  in  advance  of  the  annual  rate,  due  to  the 
fact  that  interest  is  charged  on  the  amounts  not  paid  at 
the  time  when  the  whole  premium  is  due. 


LIFE  AND  ACCIDENT  IXSUBAXCE 


111 


WHOLE    LIFE    POLICIES 

PARTICIPATING 

Age 

Payments 

20 

15 

10 

5 

Single 

FOR  Life 

Payments 

Payments 

Payments 

Payments 

Payment 

18 

$28  05 

$33  75 

$45  37 

$80  70 

$364  89 

19 

28  47 

34  24 

46  03 

8184 

369  % 

20 

28  90 

34  76 

46  71 

83  02 

375  19 

21 

$19  50 

29  35 

35  29 

47  41 

84  24 

380  58 

22 

19  93 

29  82 

35  84 

48  13 

85  50 

386  15 

23 

20  38 

30  30 

3(3  41 

48  88 

86  80 

391  89 

24 

20  86 

30  81 

37  00 

49  65 

88  14 

397  82 

25 

2135 

3133 

37  61 

50  45 

89  52 

403  93 

26 

2187 

3187 

38  24 

5128 

90  95 

410  24 

27 

22  42 

32  43 

38  90 

52  14 

92  43 

416  74 

28 

22  99 

33  01 

39  57 

53  02 

93  5K) 

423  45 

29 

23  59 

33  61 

40  28 

53  i^ 

95  53 

430  37 

30 

24  22 

»4  24 

4101 

54  89 

97  16 

437  50 

31 

24  89 

34  89 

4177 

55  87 

98  84 

444  86 

32 

25  59 

35  58 

42  55 

5()  89 

100  58 

452  44 

33 

2(5  33 

36  29 

43  37 

57  94 

102  38 

460  25 

34 

27  11 

37  03 

44  21 

59  03 

104  23 

4(xS  30 

35 

27  93 

37  80 

45  10 

60  16 

106  14 

476  58 

36 

28  80 

38  61 

46  01 

6133 

108  11 

485  12 

37 

29  72 

39  45 

46  97 

62  &1 

110  15 

493  91 

38 

30  69 

40  34 

47  96 

63  80 

112  26 

502  95 

39 

3171 

4126 

48  99 

65  10 

114  43 

512  24 

40 

32  80 

42  24 

50  07 

66  45 

116  67 

52180 

41 

33  95 

43  2(1 

5120 

67  85 

118  98 

531  ()2 

42 

35  17 

44  34 

52  38 

69  30 

12137 

54171 

43 

36  47 

45  48 

53  62 

70  82 

123  83 

552  07 

44 

37  84 

46  68 

54  92 

72  40 

126  38 

562  70 

45 

39  31 

47  95 

56  28 

74  04 

129  01 

573  59 

46 

40  86 

49  30 

57  72 

75  75 

13172 

584  76 

47 

42  52 

50  73 

59  23 

77  54 

11:^4  52 

59(5  18 

48 

44  29 

52  25 

60  82 

79  40 

137  42 

607  85 

-49 

46  17 

53  87 

62  49 

81  35 

140  41 

619  76 

50 

48  17 

55  59 

64  26 

83  38 

143  48 

63189 

51 

50  31 

57  43 

6()13 

85  50 

146  (55 

644  22 

52 

52  58 

59  38 

68  10 

87  72 

149  92 

ma  74 

53 

55  00 

6147 

70  19 

J)0  03 

153  28 

669  43 

54 

57  59 

63  71 

72  40 

92  46 

156  74 

682  28 

55 

60  34 

66  10 

74  75 

94  99 

1()0  30 

695  27 

56 

63  28 

68  6() 

77  24 

97  (>6 

l(i3  97 

708  38 

57 

66  42 

7141 

79  <K) 

100  45 

167  75 

721  m 

58 

69  78 

74  37 

82  74 

103  39 

171  ()5 

TM  91 

59 

73  37 

77  55 

85  77 

10()  50 

175  ()8 

748  28 

60 


77  20 


80  97 


89  02 


109  7' 


179  S4 


761  71 


212 


INSURANCE 


ENDOWMENT   POLICIES 

PAYMENTS   FOR    FULL  TERM. 

PARTICIPATING 

Due  i-v 

Due  IX 

Due  in 

Due  in 

Due  in 

Due  in 

Due  in 

Age 

10  Years 

15  Years 

20  Y^EARS 

25  Years 

30  Y'ears 

85  Y'ears 

40  Y^eaks 

18 

Si 02  37 

$66  25 

$48  55 

$38  22 

$3158 

$27  08 

$23  93 

19 

102  45 

66  34 

48  65 

38  32 

3170 

27  21 

24  09 

20 

102  54 

66  44 

48  75 

38  43 

3182 

27  35 

24  26 

21 

102  64 

66  54 

48  86 

38  55 

3196 

27  51 

24  44 

22 

102  73 

66  64 

48  97 

38  68 

32  10 

27  68 

24  65 

23 

102  83 

66  75 

49  09 

38  81 

32  26 

27  86 

24  86 

24 

102  94 

66  87 

49  22 

38  96 

32  42 

28  06 

25  10 

25 

103  06 

66  99 

49  36 

39  11 

32  60 

28  27 

25  36 

26 

103  18 

67  13 

49  50 

39  28 

32  79 

28  50 

25  64 

27 

103  30 

67  27 

49  66 

39  46 

33  00 

28  75 

25  95 

28 

103  44 

67  42 

49  83 

39  65 

33  23 

29  03 

26  28 

29 

103  58 

67  58 

50  00 

39  85 

33  48 

29  33 

26  65 

30 

103  74 

67  75 

50  20 

40  08 

33  75 

29  66 

27  05 

31 

103  90 

67  93 

50  41 

40  32 

34  04 

30  02 

27  48 

32 

104  08 

68  12 

50  63 

40  59 

34  37 

30  41 

27  96 

33 

104  26 

68  34 

50  87 

40  88 

34  72 

30  84 

28  48 

34 

104  46 

68  56 

5114 

4120 

35  10 

3131 

29  04 

35 

104  68 

68  81 

5143 

41  55 

35  53 

3183 

29  66 

36 

104  91 

69  07 

51  74 

41  93 

35  99 

32  39 

30  33 

37 

105  16 

(59  36 

52  09 

42  35 

36  50 

33  01 

3106 

38 

105  43 

69  68 

52  47 

42  81 

37  07 

33  ()9 

3185 

39 

105  72 

70  02 

52  88 

43  31 

37  68 

34  43 

32  71 

40 

106  04 

70  40 

53  34 

43  87 

38  36 

a5  24 

33  64 

41 

106  38 

70  82 

53  84 

44  49 

39  10 

36  12 

42 

106  76 

7127 

54  40 

45  16 

39  92 

37  09 

43 

107  18 

7178 

55  01 

45  91 

40  82 

38  14 

44 

107  64 

72  33 

55  69 

46  73 

4181 

39  29 

45 

108  14 

72  95 

56  44 

47  64 

42  90 

40  54 

46 

108  70 

73  63 

57  27 

48  64 

44  09 

47 

109  32 

74  39 

58  18 

49  75 

45  39 

48 

110  00 

75  22 

59  20 

50  97 

46  82 

49 

110  76 

76  14 

60  31 

52  31 

48  38 

50 

11159 

77  15 

6154 

53  78 

50  07 

51 

112  51 

78  27 

62  89 

55  38 

52 

113  51 

79  49 

64  38 

57  15 

53 

114  61 

80  84 

66  01 

59  07 

54 

115  82 

82  33 

67  81 

61  18 

55 

117  16 

83% 

69  78 

63  47 

56 

118  62 

85  76 

7194 

57 

120  22 

87  73 

74  31 

58 

121  98 

89  91 

76  91 

59 

123  92 

92  30 

79  75 

60 

126  05 

94  93 

82  85 

LIFE  AND  ACCIDENT  INSURANCE  213 

The  premiums  on  life  policies  are  to  be  paid  during  the  entire  life 
of  the  insured,  or  during  the  period  indicated  in  the  preceding  talkie. 
The  face  of  the  policy  is  to  be  paid  at  the  death  of  the  insured.  The 
endowment  policy  provides  for  the  payment  of  the  face  of  the  policy  in 
10,  15,  20,  25,  30,  85,  or  40  yr.  from  the  date  of  issue,  or  at  the  death 
of  the  insured  if  it  occurs  before  the  close  of  the  stated  period. 

317.  The  premiums  charged  by  the  life  insurance  com- 
panies are  determined  by  three  considerations :  (1)  the 
probability  that  the  insured  will  live  as  long  as  a  healtliy 
person  of  his  age  may  be  expected  to  live ;  (2)  the  rate  of 
interest  the  company  can  earn  on  the  premiums  paid  in ; 
(3)  the  necessary  expense  of  managing  the  company. 

In  order  to  secure  safety  of  the  policy  contract,  premiums  are  made 
higher  than  the  above  considerations  render  necessary.  The  portion 
of  the  premium  remaining  unused  at  the  end  of  any  year  may  be 
returned  to  the  policy  holder  in  the  form  of  an  annual  dividend,  or  it 
may  be  allowed  to  accunmlate  for  a  term  of  years,  called  the  accumu- 
lation period.  The  period  is  usually  10,  15,  or  20  years.  In  the  latter 
case,  no  dividend  is  paid  unless  the  policy  is  kept  in  force  to  the  end 
of  the  accumulation  period. 

The  excess  of  assets  over  liabilities  due  to  accumulated  dividends, 
interest  earned,  etc.,  forms  the  surplus  of  the  company.  The  reserve 
of  a  company  is  the  amount  held  to  meet  the  payment  of  policies 
when  due. 

Great  care  is  taken  by  life  insurance  companies  to  protect  the  in- 
sured against  forfeiture  through  nonpayment  of  premiums. 

318.  The  following  tables  illustrate  the  loan  value,  or  the  amount 
the  company  agrees  to  loan  the  insured  if  the  policy  is  assigned  to  the 
company  as  security;  the  cash  value,  or  the  amount  the  company 
agrees  to  pay  the  insured  on  surrender  of  the  policy;  the  paid-up 
policy,  or  the  face  of  a  paid-up  policy  the  company  agrees  to  exchange 
for  the  original  policy  if  surrendered ;  the  extended  insurance,  or  the 
time  the  company  will  continue  the  full  amount  of  insurance  without 
further  payment.  These  privileges  are  granted  in  consideration  of 
the  premiums  already  paid. 


214 


INSURANCE 


20-Payment  Life 


20-Year  Endowment 


Pkemiu.m  : 

Age  35 

A.       $37.80 
S.  A.     19.60 
Q.            9.98 

YEAR 

Paid 

Extended 

Cash 

Insurance 

Value 

Policy 

Years 

Days 

3 

^Al 

153  ' 

$131 

6 

194 

4 

TO 

78 

183 

8 

276 

5 

94 

105 

235 

10 

317 

6 

118 

132 

287 

12 

2t54 

7 

144 

160 

339 

14 

167 

8 

no 

189 

391 

15 

332 

9 

196 

218 

442 

17 

55 

10 

224 

249 

493 

18 

75 

11 

252 

281 

544 

19 

41 

12 

281 

313 

595 

19 

324 

13 

312 

347 

(U6 

20 

205 

14 

343 

382 

696 

21 

56 

15 

376 

418 

746 

21 

251 

16 

40S 

454 

797 

22 

69 

17 

441 

491 

847 

22 

250 

18 

476 

529 

898 

23 

76 

19 

511 

568 

948 

23 

287 

20 
25 

548 
599 

609 
666 

30 

650 

723 

Policy  full  paifl. 

Premum  : 

Age  35. 

A. 

S.A 

$51.43 
.     26.67 

(> 

13.57 

YEAR 

Loan 

Cash 
Value 

Paid 

up 

Policy 

Extended 
Insurance 

En- 
dow- 

Years 

meni 
Days            1 

3 

$82 

$92 

$147 

10 

196 

4 

118 

132 

203 

13 

84vS 

5 

155 

173 

259 

15 

0 

$41 

6 

193 

215 

314 

14 

0 

121 

7 

233 

259 

368 

13 

0 

198 

8 

274 

305 

421 

12 

0 

272 

9 

316 

352 

474 

11 

0 

343 

10 

360 

401 

525 

10 

0 

411 

11 

405 

451 

576 

9 

0 

477 

12 

453 

504 

626 

8 

0 

541 

13 

502 

558 

675 

7 

0 

602 

14 

553 

615 

724 

(3 

0 

660 

15 

606 

674 

771 

5 

0 

716 

16 

659 

733 

818 

4 

0 

781 

17 

716 

796 

865 

3 

0 

842 

18 

774 

861 

910 

2 

0 

898 

19 

835 

928 

955 

1 

0 

951 

20 

1000 

EXERCISE   73 

From  the  tables  find  the  annual  premium  required  for  : 

1.  A  life  policy  for  %  2500,  age  24. 

2.  A  ten-payment  life  policy  for  $  4000,  age  29. 

3.  A  ten-year  endowment  policy  for  %>  5000,  age  40. 

4.  A  twenty-payment  life  policy  for  $3000,  age  87. 

5.  A  twenty-year  endowment  policj^  for  $6000,  age  37. 

6.  At  age  24  Mr.  Robbins  takes  out  a  life  policy  for 
15000  ;  if  he  dies  at  the  age  of  41,  how  much  does  tlie  face 
of  the  policy  exceed  the  premiums  paid  ? 

7.  If  money  is  worth  6^  per  annum,  what  do  the 
premiums  paid  in  Ex.  6  amount  to  ?  How  much  does  tlie 
face  of  the  policy  exceed  the  amount  ? 


LIFE  AND  ACCIDENT  INSURANCE  215 

8.  At  jige  35  Mr.  Andrews  takes  out  a  -t  5000  twenty- 
payment  life  policy  ;  what  is  the  face  of  the  paid-up  life 
policy  that  will  he  given  to  him  if  he  stops  ])aying  premiums 
and  surrenders  his  ])()licy  at  age  4(j  ?  What  is  the  guar- 
anteed cash  value  of  the  policy  at  age  45  ? 

9.  At  age  31  a  man  took  out  a  ^  2500  life  policy  and  at 
age  36  a  %  1500  twenty-five-year  endowment  policy  and  a 
$  1000  twenty-year  endowment  policy.  How  much  does 
the  insurance  exceed  the  premiums  paid  if  he  dies  at  the 
age  of  43  ? 

10.  If  the  annual  dividends  on  a  twenty-payment  life 
policy,  age  35,  average  21  fo  of  the  premiums,  liow  mucli 
has  a  8  1000  policy  cost  at  the  end  of  20  years,  money  being 
worth  5  fo  ? 

Sugr/estion.  $  37.80  -  21  %  of  $  37.80  =  $29.86  =  the  average  yearly 
cost,  and  20  x  129.86  +  10  x  d%  of  20  x  i$ 29.86  =  1895.80  =  the  total 
cost. 

11.  If  dividends  are  not  paid  annually,  but  are  allowed 
to  accumulate  for  a  period  of  twenty  years  on  the  above 
twenty-payment  life  policy,  the  insured  would  be  privi- 
leged to  withdraw  the  accumulated  surplus  in  cash  and 
still  retain  a  full-paid  policy  for  i  1000  payable  at  deatli. 
Should  the  accumulated  surplus  amount  to  8  391.78  at  the 
end  of  twenty  years,  how  much  does  the  policy  cost,  money 
being  worth  5^0  ? 

12.  Mr.  Young  takes  out  a  $  5000  fifteen-payment  life 
policy  Nov.  19,  1887,  at  age  40.  In  1902,  instead  of  con- 
tinuing the  insurance,  he  surrenders  for  a  cash  value  of 
§4036.75,  which  includes  the  accumulated  dividends. 
Allowing  il5  per  annum  per  $1000  for  protection  af- 
forded, what  rate  of  interest  has  his  money  earned  in  the 
15  years  ? 


TAXES   AND   DUTIES 

319.  The  expenses  of  the  United  States  government  for 
pensions,  army  and  navy,  salaries  of  the  President,  con- 
gressmen, and  other  officials,  etc.,  amount  to  something 
over  1 1000000  a  day.  The  state  must  have  money  for 
the  care  of  the  insane,  blind,  deaf  and  dumb,  criminals ; 
for  educational  purposes,  salaries  of  state  officials,  etc. 
The  county  needs  money  for  public  buildings,  bridges, 
salaries,  educational  purposes,  etc.  The  city  and  village 
must  have  public  improvements,  fire  protection,  police, 
schools,  etc.     These  expenses  are  met  by  taxes. 

TAXES 

320.  The  expenses  for  the  support  of  the  state,  county, 
city,  etc.,  are  paid  by  taxes  on  real  estate  and  personal 
property.  In  addition  to  the  property' tax  most  states 
collect  a  poll  tax  of  from  11  to  f  3  from  each  male  citizen 
over  21  years  of  age  and  under  50. 

321.  The  rate  of  taxation  is  usually  expressed  as  a 
certain  number  of  mills  on  each  dollar,  or  as  a  certain 
number  of  cents  on  each  $  100  of  valuation. 

EXERCISE   74 

1.  Tlie  valuation  of  the  property  of  a  certain  county  is 
$  7500000.  If  the  general  state  tax  and  tlie  general  county 
tax  are  each   60  ct.   on  each  $  100  and   in   addition  the 

216 


TAXES  AND  DUTIES  217 

bridge  tax  is  40  ct.  and  the  school  tax  30  ct.,  what  is  tlie 
total  tax  of  the  county  and  what  is  the  amount  set  aside 
for  each  of  the  above  purposes  ? 

2.  What  are  the  taxes  of  a  man  who  owns  160  acres  of 
land  in  the  above  county  worth  ^  60  an  acre  and  assessed 
at  I   of   its   value,  and  personal  property   amounting  to 

11850? 

3.  The  total  assessed  value  of  property  in  Michigan  in 
1901  was  '^1578100000.  What  amount  did  the  State 
University  receive  in  1903  from  a  |  of  a  mill  tax  ? 

4.  How  much  of  this  tax  did  a  farmer  have  to  pay  who 
owns  200  acres  of  land  valued  at  ^  lb  an  acre  and  assessed 
at  I  of  its  value  ? 

5.  A  certain  city  is  bonded  for  $  6000 ;  its  taxable 
property  is  valued  at  8  7500000.  How  much  of  the  above 
bonded  indebtedness  does  a  man  worth  %>  10000  j)ay  ? 

6.  Suppose  the  above  city  wishes  to  build  a  high  school 
building  valued  at  $  50000,  what  will  be  the  tax  on  each 
1 100  ? 

7.  The  taxable  property  of  a  certain  county  is 
$125000000.  What  will  be  the  tax  on  each  $>100  to 
build  a  courthouse  worth  ^  90000  ? 

8.  The  Michigan  State  Normal  College  received  from 
the  state,  in  1903,  -$103200.  How  much  of  this  did  a 
man  pay  who  owns  -f  7500  worth  of  taxable  property,  the 
state  having  property  listed  at  $  1578100000  ? 

9.  The  assessed  value  of  a  town  is  -^S^  250000  and  tlie 
amount  of  tax  to  be  raised  is  -$3500;  Avhat  is  the  rate  of 
taxation  ? 


218  TAXES  AND  DUTIES 

DUTIES 

322.  The  income  for  the  support  of  the  national  gov- 
ernment is  derived  largely  from  custom  revenue  (tariff  or 
duty  on  imports,  collected  at  customhouses  established  by 
the  government  at  ports  of  entry),  and  internal  revenue 

(taxes  on  spirits,  tobacco,  etc.). 

323.  ^lerchandise  brought  into  the  country  is  subject 
to  ad  valorem  duty  (a  certain  per  cent  of  the  cost  of  the 
goods),  specific  duty  (a  certain  amount  of  weight,  number, 
or  measure,  without  regard  to  value),  or  both  ad  valorem 
and  specific  duty.      Some  goods  are  admitted  duty  free. 

Illustrations.  Cut  glass  and  laces  pay  an  ad  valorem  duty  of  60%. 
Machinery  pays  45%.  Tin  plates  pay  a  specific  dutj'-  of  li  ct.  per 
pound,  horses  valued  at  $150  or  less  pay  $30,  and  wheat  pays  25  ct. 
per  bushel.  Cigars  pay  a  duty  of  $4.50  per  pound  and  25%,  and  lead 
pencils  pay  45  ct.  per  gross  and  25%.  Books  published  in  foreign  lan- 
guages are  admitted  duty  free. 

EXERCISE   75 

1.  What  will  be  the  duty  on  1  T.  4  cwt.  of  tin  plate  ? 

2.  What  will  be  the  duty  on  20  gross  of  lead  pencils? 

3.  What  is  the  cost  per  gross  of  lead  pencils  on  Avhich 
the  two  rates  of  duty  are  equal  ? 

4.  The  duty  on  ready-made  clothing  is  50%.  What 
is  the  duty  on  -16000  worth? 

5.  If  the  duty  on  linen  collars  and  cuffs  is  40  ct.  per 
dozen  and  20%,  wliat  is  the  duty  on  10  doz.  collars  at 
75  ct.  a  dozen  and  10  pairs  of  cuffs  at  25  ct.  a  pair  ? 

6.  What  is  the  duty  at  50%  on  500  doz.  kid  gloves  at 
75  francs  a  dozen  ? 

7.  Find  the  duty  on  an  importation  of  ^750  8s.  4:d. 
worth  of  English  crockery  at  40%. 


THE   PROGRESSIONS 

324.  By  a  series  is  meant  a  succession  of  terms  formed 
according  to  some  common  law. 

325.  An  arithmetical  progression  (A.  P.)  is  a  series  in 
which  each  term  tlifl'ers  from  the  preceding  by  a  constant 
quantity  called  the  common  difference. 

Thus,  2,  5,  8,  11,  •••,  and  1.5,  10,  5,  0,  -  5,  -  10,  •••,  are  arithmetical 
progressions.  In  the  first,  3  is  the  common  difference  and  is  added 
to  each  term  to  form  the  next;  in  the  second,  —5  is  the  common 
difference  and  is  added  to  each  term  to  form  the  next.    . 

326.  A  geometrical  progression  (G.  P.)  is  a  series  in  which 
each  term  differs  from  the  preceding  by  a  constant  multi- 
plier called  the  ratio. 

Thus,  2,  4,  8,  16,  ••.,  and  18,  -  6,  2,  -  |,  |,  ...,  are  geometrical  pro- 
gressions, the  ratios  being  respectively  2  and  —  -|. 

327.  Last  Term.  If  a  is  the  first  term,  I  the  last  term, 
d  tlie  common  difference,  r  the  ratio,  and  n  the  number  of 
terms,  we  have  from  the  definitions,  — 

1st  2(1  3d  «th 

A.  P.         n         ('/  +  '/)  {a +  2(1)  ...         a+{n-\)d 

G.  P.         a  or  ar^ 


•  n—i 


.'.  the  formulas  for  tlie  last  of  n  terms  are: 

A.  P.     l  =  a  +  (n  -  l)d. 

G.  P.     1  =  a/"'-i. 
219 


220  THE  PROGRESSIONS 

Mr.  Find  the  last  term  in  an  xV.  P.  in  which  the  first 
term  is  10,  the  common  difference  4,  and  the  number  of 
terms  12. 

Solution.     I  =  a-\-  (n-  l)d  =  10  +  (12  -  1)  x  4  =  54. 

Mx.  Find  the  hist  term  in  a  G.  P.  in  which  the  first 
term  is  2,  tlie  common  ratio  3,  and  the  number  of  terms  5. 

Solution.     I  =  a?-''-'^  =2x3^  =  162. 


328.    Sum  of  Series. 

A.  P.  Take  the  series  o,  5,  7,  9,  11,  in  which  «  =  3,  d  =  2,  I  =  11, 
and  the  snni  (S)=  35. 

Then         ^^=    3+    (3  +  2)+   (3  +  4)  +  (3  +  G)    +(3  +  8), 

and  in  reverse  order 

^^  z.  11  +  (11  -  2)  +  (11  -  4)  +  (11  -  G)  +  (11  -  8). 

Adding  and  canceling  tlie  2,  4.  6,  and  8, 

2  ^^  =  (3 +  11) +  (3 +  11) +  (3 +  11) +  (3 +11)  + (3 +  11) 

3=5(3  +  11). 

.-.  5:  =  |(3  +  11).=:35, 

or  the  sum  of  the  series  equals  one  half  of  the  number  of  teryns  times  the 
sum  of  the  first  and  last  terms. 

Take  the  general  series  a,  a  +  d,  a  +  2  d,  •••,  a  -\-  {n  —  l)d.  In  this 
series  it  will  be  noticed  that  each  term  is  formed  by  adding  to  the  first 
term  the  common  difference  mnltiplied  by  the  number  of  the  term 
less  one. 

Then         5  =  rt  +  (rt  +  </)  +  (a  +  2  d)  +  •..(/  -  d)  +  /, 
and  in  reverse  order 

S  =  l+  (l-d)+  (/_o,/)  +  ...(rt+,/)  +  a. 


THE  PROGRESSIONS  221 

Adding  and  canceling  the  (Vs. 

2  S  =  a  +  I  -\-  a  -\- 1 -{-  a  +  I  +  '" a  i-  I  -^  a  +  I  =  n(a  -\-  i). 

G.  P.  Take  tlie  series  2,  G,  18,  54,  102,  in  wliich  a  =  2,  r  =  ;5,  1=  102, 
n  =  5,  and  S  =  242. 

Then  5  =  2  +  2x3  +  2x  3H2  x  8-H2  x  31 

Multii)ly ing  by  3,  3  ,S:  =         2x3  +  2  x  3-^  +  2  x  33  +  2  x  3^  +  2  x  3^. 

.  Subtracting  and  canceling  common  terms, 

5  -  3  5  =  2  -  2  X  35. 

.».  ^^  =  - — =^^^  =  242, 
1-3 

ov  the  sum  of  the  series  equals  the  frst  term  minus  the  Jirst  term  times  the 
ratio  raised  to  a  power  equal  to  the  number  of  terms  divided  by  one  minus 
the  ratio. 

Take  the  general  series  a,  ar,  ar^,  •••,  ar^-i.  In  this  series  it  will  be 
noticed  that  each  term  is  formed  by  multiplying  the  first  term  by  the 
ratio  raised  to  a  power  one  less  than  the  number  of  the  terra. 

S  =  a  +  ar  -\-  ar-  +  ar^  -f h  «?•"-!. 

Multiplying  by  r,     rS  =         ar  +  «;•-  +  ar^  + h  ar"  -^  +  ar'K 

Subtracting,  S  —  rS  =  a  —  ar"". 

^  _  a  —  ar^ 
1-r  ' 

■  Ex.     Find  the   sum  of  the  first  8  terms  in  an  A.  P. 
when  the  first  term  is  5  and  the  common  difference  is  3. 

Solution.     Since  5  —  -^ — ^t_i  and  /  =«  +  {n  —  1)^/, 

.-.  5  =  'i  [2  a  +  (n  -  1)^/]  =  4  (10  +  7x3)^  124. 

Ex.     Sum  to  6  terms  the  series  2  +  6  +  18  -f  •••. 
-  2  X  36 


—  70 


1-3 


728. 


222  THE  PROGRESSIONS 

329.   Infinite  Series.     Writing  the  formula 

r,      a  —  ar^  a  ar^ 

o  =  — = , 

1  —  r        1  —  r      1  —  r 

we  see  tliat  wlien  r  is  a  proper  fraction  and  n  becomes 
large,  ar"^  becomes  small.     If  we  make  n  sufficiently  large, 

ar^  and  hence  :; will  approach  as  near  to  zero  as  we 

1-^                              ...                   a 
please,  and  hence,  when  n  becomes  infinite,  S=^ -• 

Ex.     Sum  to  infinity  the  series  l  +  2  +  i  +  i+""* 
1 


Solution.     S 


1-i 


330.   Circulating    Decimals.      \  =  0.3333  •••    and    g^  = 

0.189189  •••.  In  the  first  case  3  is  repeated  indefinitely, 
and  in  the  second  case  the  digits  189  are  repeated  indefi- 
nitely in  the  same  order.  Such  decimals  are  called  circu- 
lating decimals.  The  repeating  figures  are  called  the 
repetend.  A  circulating  decimal  is  expressed  by  writing 
the  repetend  once  and  placing  a  dot  over  the  first  and  the 
last  figure  of  the  part  repeated. 

Thus,  0.333  ...  =  0..3  and  0.189189  ...  =  0.189. 

Ex.     Reduce  0.3  to  an  equivalent  common  fraction. 
Solution.     0.3  =  /o  +  ito  +  iwoo  -f-  ".  =  a  G.  P.  with  the  first  term 

3_ 

=  1^0,  and  the  ratio  =  i^.     .-.  S  =     ^"      =  \. 

EXERCISE   76 

1.  Find  the  12th  term  of  the  series  5,  7,  9,  •••. 

2.  Find  the  7th  term  of  the  series  J,  J,  ^-,  -••. 

3.  Find  the  sum  of  9  terms  of  g  +  J  +  -2  +  ■'' 


THE   PROGRESSIONS  223 

4.  If  a  bod}"  falls  l<>j^2  ^^-  ^^^^  ^'^^'^^  second,  :]  times  as 
far  the  next  second,  5  times  as  far  the  third  second,  and 
so  on,  how  far  will  it  fall  in  the  twelfth  second?  How 
far  will  it  fall  in  12  sec.  ? 

5.  Find  the  8th  term  in  the  series  1,  \,  J,  •••. 

6.  Kind  the  snm  of  1  +  }^  +  i  +  •■•  to  infinity. 

7.  Plnd  the  7th  term  in  the  series  4,  —2,  1,  •••. 

8.  Find  the  valne  of  0.423. 

9.  Find  the  5th  and  9th  terms  of  the  series  3,  6,  12,  •••. 

10.  Find  the  9th  term  of  the  series  g^^,  3^2,  iV'  *"• 

11.  Snm  to  5  terms  the  series  9,  —6,  4,  •••. 

12.  Find  the  value  of  0.2:   0.28;   0.24;   1.7145. 

13.  Find  the  sum  of  3  +  0.3  +  0.03  H to  infinity. 

14.  Find  the  sum  of  the  first  25  odd  numbers  ;  the  first 
25  even  numbers. 

15.  What  is  the  distance  passed  through  by  a  ball 
before  it  comes  to  rest,  if  it  falls  from  a  height  of  40  ft. 
and  rebounds  half  the  distance  at  each  fall  ? 


LOGARITHMS 

331.  Early  in  tlie  seventeenth  century,  John  Napier,  a 
Scotchman,  invented  logarithms,  by  the  use  of  which  the 
arithmetical  processes  of  multiplication,  division,  evolution 
and  involution  are  greatly  abridged. 


1 

0 

2 

1 

4 

2 

8 

3 

16 

4 

32 

5 

64 

6 

128 

7 

256 

8 

512 

9 

1024 

10 

2048 

11 

4096 

12 

8192 

13 

16384 

14 

32768 

15 

65536 

16 

131072 

17 

262144 

18 

524288 

19 

1048576 

20 

332.  Many  simple  arithmetical  operations 
can  be  performed  by  the  use  of  two  columns 
of  numbers,  as  given  in  the  annexed  table. 

The  left-hand  column  is  formed  by  writing  unity  at 
the  top  and  doubling  each  number  to  get  the  next.  The 
right-hand  column  is  formed  by  writing  opposite  each 
power  of  2,  the  index  of  the  power.  Thus  512  =  2^,  the 
number  opposite  512  indicating  the  power  of  2  used  to 
produce  512. 

Ex.  1.    Multiply  4096  by  64. 

From  the  table  4096  =  21^  and  64  =  2«. 

.-.  4096  X  64  =  212  ^  2^  =  212+6  ^  ois  ^  262144  (from 
table). 

The  student  should  notice  that  the  simple  oj)eration 
of  addition  is  substituted  for  multiplication,  the  product 
being  found  in  the  left-hand  column  opposite  18,  the 
sum  of  12  and  6. 


Ex.  2.    Divide  1048576  by  2048. 

1048576  --  2048  =  2'^  -  2"  =  220-11  =  2^ 
place  of  division) . 

224 


512  (subtraction  takes  the 


LOGARITHMS  225 


Rr.  3.    Find  sy;J2768. 

\/;32708  =  v^L>i5  =  2V  =  23  =  8  (division  takes  the  place  of  evolu- 
tion). 

In  the  preceding  table  tlie  niind)ers  in  the  right-hand 
column  are  called  the  logarithms  of  the  corresponding 
numbers  in  the  left-hand  column.  2  is  called  the  base  of 
this  system.  Tlierefore,  the  logarithm  of  a  number  is  the 
exponent  by  tvMch  the  base  is  affected  to  j^Toduce  the  number. 

333.  Any  other  base  than  2  might  have  been  used  and 
columns  similar  to  the  above  formed.  In  practice  10  is 
always  taken  as  the  base  and  the  logarithms  are  called 
common  logarithms  in  distinction  from  the  natural  loga- 
rithm, of  which  the  base  is  2.71828.  Common  logarithms 
are  indices^  positive  or  7iegative^  of  the  poiver  o/  10. 

From  the  definition  of  common  logarithms,  it  follows  that  since 
100=1,         log  1  =  0. 
101  ^  10,       log  10  =  1. 
10-2  =  100,     log  100  =  2. 
103  =  1000,  log  1000  =  3. 
etc. 

334.  Since  most  numbers  are  not  exact  powers  of  10, 
logarithms  will  in  general  consist  of  an  integral  and  deci- 
mal part.  Thus,  since  log  100  =  2  and  log  1000  =  3,  the 
logarithms  of  numbers  between  100  and  1000  will  lie 
between  2  and  3,  or  will  be  2+  a  fraction.  Also  since 
log  0.01  =  -  2  and  log  0.001  =  -  3,  the  logarithms  of  all 
numbers  between  0.01  and  0.001  will  lie  between  —  2  and 
—  3  or  will  be  —  3  -f  ^  fraction.  The  integral  part  of  the 
logarithm  is  called  the  characteristic  and  the  decimal  part 
the  mantissa. 

LYMAN 'S  AT)V.    AR.  —  15 


10-1  ^  0.1, 

log  0.1  =  -  1. 

10-2  =  0.01, 

log  0.01  =  -  2. 

10-3  ^  0.001, 

log  0.001  =  -  3. 

10-4  =  0.0001, 

,  log  0.0001  =  -  4. 

etc. 

226  L0GARITH2IS 

335.  The  characteristic  of  the  logarithm  of  a  number  is 
independent  of  the  digits  composing  the  number,  but 
depends  on  the  position  of  the  decimal  point.  Charac- 
teristics, therefore,  are  not  given  in  the  tables.  Thus, 
since  246  lies  between  100  and  1000,  log  246  will  lie 
between  2  and  3,  or  will  be  2  +  a  fraction.  Again 
since  0.0024  lies  between  0.001  and  0.01,  its  logarithm 
lies  between  —3  and  —2,  or  log  0.0024  =  — 3 -f  a 
fraction. 

336.  From  the  above  illustrations  it  readily  appears 
that  the  characteristic  of  the  logarithm  of  a  nuynber^  partly 
or  ivholly  integral^  is  zero  or  positive  and  one  less  than  the 
number  of  figures  in  the  integral  part. 

337.  The  characteristic  of  the  logarithm  of  a  pure  deciynal 
is  negative  and  one  more  than  the  number  of  zeros  preceding 
the  first  significant  figure. 

EXERCISE   77 

1.  Determine  the  characteristic  of  the  logarithm  of  2  ; 
526;   75.34;  0.0005;   300.002;   0.05743. 

2.  If  log  787  =  2.8960,  what  are  the  logarithms  of  7.87, 

0.0787,  78700,  78.7? 

338.  The  mantissa  of  the  logarithm  of  a  number  is 
independent  of  the  position  of  the  decimal  point,  but 
depends  on  the  digits  composing  the  number.  Mantissas 
are  always  positive  and  are  found  in  the  tables,  for  mov- 
ing the  decimal  point  is  equivalent  to  multiplying  the 
number  by  some  integral  power  of  10,  and  therefore  adds 
to  or  subtracts  from  the  logarithm  an  integer. 


LOGARITHMS  227 

Thus,       log  76.42  =  log  76.42, 

log  764.2  =  log  76.42  x  10  =  log  76.42  +  1, 
log  7642  =  log  76.42  x  10'^  =  log  7(5.42  +  2, 
log  7.642  =  log  76.42  x  lO-^  =  log  76.42  +  ( - 1 ). 

So  that  the  mantissas  of  all  numbers  composed  of  the  digits  7642 
in  that  order  are  the  same,  since  moving  the  decimal  point  affects  the 
characteristic  alone. 

Log  0.0063  is  never  written  -  3  +  7993,  but  3.7993. 
The  minus  sign  is  written  above  to  indicate  that  the 
characteristic  alone  is  negative.  To  avoid  negative 
characteristics  10  is  added  and  subtracted.  Thus,  3.7993 
=  7.7993  -  10. 

339.  The  principles  used  in  working  with  logarithms 
ai-e  as  follows  : 

I.  The  logarithm  of  a  product  equals  the  sum  of  the 
logarithms  of  the  factors. 

II.  The  logarithm  of  a  cpwtient  equals  the  logarithm  of 
the  dividend  minus  the  logarithjii  of  the  divisor. 

III.  The  logarithm  of  a  poiver  equcds  the  index  of  the 
power  times  the  logarithm  of  the  number. 

IV.  The  logarithm  of  a  root  equcds  the  logarithm  of  the 
number  divided  by  the  index  of  the  root. 

For  let  10^  =  n  and  10^  =  m, 

then  log  n  =  X  and  log  m  =  y. 

Therefore,  since  mn  =  10-^+J', 

log  nni  =  X  -\-  y  =  log  ;i  +  log  ??i; 
and      ■  71  -^  m  =  10'-^, 

then  log—  =x  —  y  —  log  n  —  log  m. 

m 


228 

LOGARITHMS^ 

Also 

rf  =  (100 "  =  10'% 

then 

log  n'"  =  rx  =  r  log  n. 

Finally 

y/n  =  </l{)^  =  lOs 

then 

logv^=-  =  -  logn. 

EXERCISE   78 

Given  log  2  =  0.3010,  log  3  =  0.4771,  log  5  =  0.6990,  find: 

1.  log  6.  3.  log  52.  5.  log  0.18.        7.  logf. 

2.  log  15.       4.  logVlS.        6.  log  7.5.  8.  log  32x53. 
9.  Find  the  number  of  digits  in  SO^^ ;  in  25^^. 

USE  OF  TABLES 

340.  In  the  tables  here  given  the  mantissas  are  found 
correct  to  but  four  decimal  places.  By  using  these  tables 
results  can  generally  be  relied  upon  as  correct  to  3  figures 
and  usually  to  4.  If  a  greater  degree  of  accuracy  is 
required,  five-place  or  even  seven-place  tables  must  be 
used. 

341.  To  find  the  logarithm  of  a  given  number. 

Write  the  characteristic  before  looking  in  the  tables  for 
the  mantissa. 

Find  the  mantissa  in  the  tables. 

(1)  When  the  number  consists  of  not  more  than  three 
figures. 

In  the  column  N,  at  the  left-hand  side  of  the  page, 
find  the  first  two  figures  of  the  number.     In  the  row  N, 


USE  OF  TABLES  229 

at  the  top  or  bottom  of  the  page,  as  convenient,  find  the 
third  figure.  Tlie  mantissa  of  the  number  will  be  found 
at  the  intersection  of  the  row  containing  tlie  first  two 
figures  and  the  column  containing  the  third  figure. 

Ex.   Find  log  384. 

The  characteristic  is  2  (WhyV).  Tii  the  eoluiiin  N"  find  :>8  and  in 
row  N  find  4.  The  mantissa  5843  will  be  found  at  the  intersection  of 
the  row  38  and  column  4. 

.-.  log  384  =  2.5843. 

What  is  log  3.84  ?  log  38.4  ?  log  0.0384  ? 

(2)    When  the  number  consists  of  more  than  three  figures. 

Find  as  above  the  mantissa  of  the  logarithm  of  the 
number  consisting  of  the  first  three  figures.  To  correct 
for  the  remaining  figures  interpolate  by  assuming  that^  for 
differences  small  as  compared  ivith  the  numbers.,  the  differ- 
ences betiveen  numbers  are  proportional  to  the  differences 
between  their  logarithms.  This  statement  is  only  approxi- 
mately true,  but  its  .use  leads  to  results  accurate  enough 
for  ordinary  computations. 

Ux.  Find  log  3847. 

iVIantissa  of  log  3850  =     5855. 
Mantissa  of  log  3840  =     5843. 

10      0.0012. 
Mantissa  of  log  3847  =  5813  +  j\  of  0.0012  =  5851. 

The  difference  between  3840  and  3850  is  10,  the  difference  between 
the  mantissas  of  their  logarithms  (5855  —  5843)  is  0.0012.  Assuming 
that  each  increase  of  1  unit  between  3840  and  3850  produces  an  in- 
crease of  1  tenth  of  the  difference  in  the  mantissas,  the  addition  for 
3847  will  be  7  tenths  of  0.0012  or  0.00081.  5813  +  0.00084  =  5851. 
Therefore,  the  mantissa  of  log  3847  =  5851. 


230 


LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0:334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0(507 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

09;34 

0969 

1004 

1038 

1072 

110(5 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1:399 

1430 

14 
15 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

1() 

2041 

2068 

2095 

21 '>2 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

24:30 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 
20 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

33(i5 

3385 

3404 

22 

3124 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 
25 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

41:33 

26 

4150 

4106 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4:346 

4362 

4378 

4393 

4409 

4425 

4440 

445(5 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4(509 

29 
30 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4{H)0 

31 

4914 

4928 

4;>42 

4955 

4969 

4983 

4997 

5011 

5024 

50:58 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 
35 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36. 

5563 

5575 

5587 

5599 

5011 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

'5752 

5763 

5775 

578(5 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 
40 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

(3010 

6021 

6031 

6042 

6053 

6004 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

(5222 

42 

6232 

(5243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

(5:i25 

43 

6335 

6345 

6355 

6365 

6375 

6:385 

6395 

6405 

6415 

6425 

44 
45 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6.-22 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

()637 

6(546 

6656 

6665 

6675 

6684 

6693 

6702 

(5712 

47 

6721 

6730 

<5739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6312 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 
50 

6902 

6911 

6920 

6928 

6937 

(5946 

6955 

6964 

6972 

6i)81 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

70(57 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

72<»2 

7300 

7:308 

731(5 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

739(3 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

USE  OF  TABLES 


231 


N 

0 

1 

2 

3 

4: 

5 

6 

7 

s 

{) 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

74(56 

7474 

")(; 

7482 

741M) 

7497 

7505 

7513 

7520 

7528 

75;;() 

754:5 

7551 

57 

7559 

75()(; 

7574 

7582 

7589 

7597 

7(i()4 

7(512 

7619 

7(527 

58 

Hi'M 

7642 

7()49 

7657 

7664 

7672 

7679 

7686 

7(594 

7701 

5!) 
60 

7709 

7716 

7723 

7731 

773.8 

7745 

7752 

77(50 

77(57 

7774 

7782 

7789 

771KJ 

7803 

7810 

7818 

7825 

7832 

7839 

784(5 

(il 

7853 

7860 

78<i8 

7875 

7882 

7889 

789(i 

7903 

7910 

7917 

()2 

7924 

7931 

7938 

7945 

7952 

7959 

7!Mi6 

7973 

7980 

7987 

(i;i 

7993 

8(M)0 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8(J55 

()4 
65 

80G2 

8069 

8075 

8082 

8089 

809() 

8102 

8109 

8116 

8122 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

817(5 

8182 

8189 

(i(> 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

<!7 

8201 

82()7 

8274 

8280 

8287 

8293 

8299 

830(5 

8312 

8319 

()S 

8325 

8331 

8338 

8:H4 

8351 

8357 

Hdd'S 

8370 

8376 

8;)82 

70 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

850(5 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

a5(51 

85(57 

72 

8573 

8579 

8585 

8591 

8597 

8(i03 

8609 

8(515 

8621 

8627 

7H 

8()33 

8639 

8()45 

8651 

8(J57 

86(13 

8669 

8(575 

8(581 

8686 

74 
75 

8092 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

7() 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8«.K)0 

8965 

8971 

79 
80 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

91(55 

9170 

9175 

9180 

9186 

8o 

9191 

91<)() 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 
85 

9243 

9248 

9253 

9258 

9263 

92(59 

9274 

9279 

9284 

1)289 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

8() 

9345 

9350 

9355- 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

94<i0 

9465 

94(>9 

9474 

9479 

9484 

9489 

81) 
90 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

9542 

9547 

9552 

9557 

9562 

95(56 

9571 

9576 

9581 

9586 

1)1 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9(524 

9628 

9(533 

92 

9638 

9()43 

9647 

9652 

9657 

9()61 

9(56(5 

9671 

9675 

9680 

93 

9(585 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 
95 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

982,^ 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

98(53 

97 

9868 

9872 

9877 

9881 

9886 

9890 

981U 

9899 

9903 

9W8 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

99<)6 

X 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

232  LOGARITHMS 

EXERCISE   79 

1.  Find  log  1845. 

2.  Find  log  6.897. 

3.  Find  log  0.04253. 

342.  To  find  the  number  corresponding  to  a  given 
logarithm. 

The  number  corresponding  to  a  logarithm  is  called  the 
antilogarithm. 

The  characteristic  determines  the  position  of  the  deci- 
mal point. 

(1)  If  the  mantissa  is  found  iyi  the  tables^  the  number  is 
found  at  once. 

Ex.  1.    Find  antilog  3.5877. 

The  mantissa  is  found  at  the  intersection  of  column  7  and  row  38. 
.-.  antilog  3.5877  =  3870. 

(2)  If  the  exact  mantissa  is  not  found  in  the  tables^ 
the  first  three  figures  of  the  corresponding  number  can 
be  found  and  to  them  can  be  annexed  figures  found  by 
interpolation. 

Ex,  2.    Find  antilog  3.5882. 

log  3880  =  3.5888  log  required  number  =  3.5882 

log  3870  =  3.5877  log  3870  =  3..5877 

10      0.0011  log  req.  no.  -  log  3870  =  0.0005 

3870  +  f  A  of  lo]  =  3874.54+. 

The  two  mantissas  in  the  table  nearest  to  the  given  mantissa  are 
5888  and  5877  differing  by  0.0011.  Their  corresponding  numbers, 
since  the  characteristic  is  3,  are  3880  and  3870,  differing  by  10.     The 


USE  OF  TABLES  233 

diiference  between  the  smaller  mantissa  5877  and  the  required  man- 
tissa 5882  is  0.0005.  Since  an  increase  of  11  ten  thousandths  in 
mantissas  corresponds  to  an  increase  of  10  in  the  numbers,  an  increase 
of  5  ten  thousandths  in  mantissas  may  be  assumed  to  corres])oiid  to 
an  increase  oi'{\  of  10  in  the  numbers.  Therefore  the  number  is 
3870  +(r\  of  10)  =  3874.54+. 

EXERCISE   80 

1.  Find  antilog  2.9445;  aiitilog  2.40G5. 

2.  Find  antilog  1.6527  ;  antilog  3.7779. 

3.  Find  antilog  1.9994;  antilog  0.7320, 

343.  Tlie  cologarithm  of  a  number  is  the  logaritlnn  of 
its  reciprocal.  The  cologarithm  of  100  equals  the  loga- 
rithm of  Y^Q-,  i.e.  —  2.  As  the  cologarithm  of  a  number 
equals  the  logarithm  with  its  sign  changed,  adding  the 
cologarithm  will  give  the  same  result  as  subtracting  the 
logarithm.     The  former  is  sometimes  more  convenient. 

Since  log  1  =  0, .-.  log  _  =  log  1  —  log  n  =0  —  log  n, 

n 

therefore  colog  n  =  —  log  n. 

To  avoid  negative  results  it  is  often  more  convenient  to  add  and 
subtract  10. 

Then  colog  n  =  10  —  log  n  —  10. 

Ex.  1.     Find  colog  47.3. 

log  1  =  10.0000  -  10 

log  47.3  =  1.6749 

colog  47.3  =  8.3251  -  10 

In  subtracting  1.6740  or  any  other  logarithm  from  10.  the  result 
may  be  obtained  mentally  by  subtracting  the  right  hand  figure  from 
10  and  all  the  others  from  9. 


234  LOGARITHMS 

452  X  23 


Ex.  2.     Find  the  value  of 


5871  X  29 


log  ^^-  ^  "-^  =  log  452  +  log  23  -  log  5371  -  log  29 
^  5371  X  29  ^  "^  "^  ^ 

=  log  452  +  log  23  +  colog  5371  +  colog  29 
log  452  =  2.6551 
log  23  =  1.3617 

colog  5371  =6.2699 -10 

colog  29  =  8.5376  -  10 

log  0.066728+=  8.8243  -  10 

Therefore  '^''^"  ^  "'^  =  0.06672S+. 

5371  X  29 

Ux.  3.    Find  log  50'. 

log  50?  =  I  log  50 

log  50  =  1.6990 
f  log  50  =  I  of  1.6990  =  1.2742 
1.2742  =  log  18.8 
.♦.  50i-  =  18.8. 

EXERCISE   81 

Find  the  value  of  : 

1.    (5x4^7)*.  6.    0.0625 -^  0.25. 

1  ^     31  X  47  X  53 

2-    225'  "    29x43  x50' 


3/23  X  30  3        ;<i21  x  4325 

^      12      '  '    ^        729 


^     3.14  X  56.7  9.  vV  X  10.16. 

29         • 

10.  7r2;    -, 

5.    (0.625)^^.  TT 


EXERCISES   FOR   REVIEW 

In  connection  with  each  exercise  the  student  sliouhl 
review  all  principles  involved.  The  following  list  will 
then  furnish  a  complete  review  of  the  book. 

1.  What  are  the  various  names  given  to  the  symbol  0  ? 

2.  Read  the  numbers  200,  0.02;  100.045,0.145. 

3.  Solve  4672  -  2184  +  7635  +  2377  -  8432  by  adding 
the  proper  arithmetical  complements  and  subtracting  the 
proper  powers  of  10. 

4.  Multiply  5280  by  25;  by  16f . 

5.  Multiply  1760  by  9;  by  11 ;  by  81  ;  by  16. 

6.  Multiply  4763  by  998. 

7.  Multiply  4634  by  4168. 

8.  Multiply  746  by  18. 

9.  Show  that  to  multiply  a  number  by  1.5  is  the  same 
as  to  add  ^~  of  the  number  to  the  multiplicand. 

10.  Show  that  to  divide  a  number  by  112^  is  the  same 
as  to  move  the  decimal  point  two  places  to  the  left  and 
subtract  ^  of  the  number. 

11.  Form  a  table  of  multiples  of  the  multiplier  and  mul- 
tiply (a)  7461,  (6)  3465,  (c)  761,  (d)  98723,  (e^)  1846,  each 
by  3762.     Also  find  each  product  by  using  logarithms. 

12.  Form  a  table  of  multiples  of  the  divisor  and  divide 
(«)  7346,  (b)  5280,  (c)  8976,  (d)  4284,  each  by  361.  Also 
find  each  quotient  by  using  logarithms. 


2B6  EXERCISES  FOR  REVIEW 

13.  Show  that  every  number  divisible  by  4  is  the  sum 
of  two  consecutive  odd  numbers. 

14.  Show  that  the  sum  and  difference  of  two  odd  num- 
bers are  always  even. 

15.  Prove  that  the  difference  between  a  number  and  the 
number  formed  by  writing  its  digits  in  reverse  order  is 
divisible  by  9. 

16.  Perform  the  following  operations  and  check  by  cast- 
ing out  the  9's  :  86942  x  763 ;  46342  -  216  ;  842  x  21.34 ; 
987.4-3.1416.* 

17.  Find  the  quotient  of  764321  divided  by  2136  correct 
to  four  significant  figures. 

18.  Find  the  quotient  of  76.421  divided  by  3.1416  cor- 
rect to  0.01. 

19.  Multiply  5276  by  121  and  divide  the  result  by  331. 

20.  Prove  that  a  number  is  divisible  by  4  if  the  units' 
digit  minus  twice  the  tens'  digit  is  divisible  by  4. 

21.  When  it  is  10  p.m.  Sunday,  Feb.  15,  at  GreeuAvich, 
what  time  and  date  is  it  at  165°  W.? 

22.  Suppose  a  transport  returns  troops  from  Manila 
starting  July  4,  reaching  San  Francisco  35  da.  later  ;  what 
is  the  date  ? 

23.  27  is  composed  of  16  and  11 ;  write  all  of  the  other 
two  numbers  that  make  up  27. 

24.  Reduce  43132^  to  the  decimal  scale. 

25.  What  methods  did  the  ancient  Babylonians,  Egyp- 
tians, Greeks  and  Romans  adopt  to  represent  numbers  ? 
Were  these  characters  ever  employed  as  instruments  of 
calculation  ? 

*  Perform  also  by  logarithms. 


EXERCISES  FOR   REVIEW  237 

26.  From  what  source  was  the  decimal  system  of  iKjta- 
tion  with  its  9  digits  derived  ? 

27.  Exphdu  clearly  the  difference  between  tlie  intrinsic 
value  and  the  local  value  of  the  \)  digits. 

28.  In  the  decimal  scale  exphiin  why  tlie  lunnljcr  of 
characters  used  cannot  be  more  nor  less  tlian  10. 

29.  What  is  the  difference  between  the  sum  of  4G23, 
256,  145231,  7649,  and  a  million  ? 

30.  Find  the  excess  of  864213  over  634795  by  means  of 
arithmetical  complements. 

31.  Multiply  37635  by  648,  using  but  two  partial  prod- 
ucts. 

32.  Prove  that  any  number  composed  of  three  consecu- 
tive figures  is  divisible  by  3. 

33.  Find  the  sum  and  difference  of  6523  and  5436  in  the 
scale  of  8. 

34.  Multiply  529  t  by  1903  in  the  scale  of  12. 

35.  Divide  4234  by  213  in  the  scale  of  5. 

36.  What  weights  must  be  selected  out  of  1,  3,  9,  27, 
81,  etc.,  pounds  to  weigh  1907  lb.  ? 

37.  A  carriage  wheel  revolves  2  times  in  going  25  ft. ; 
how  many  times  will  it  revolve  in  going  a  mile  ? 

38.  How  mucli  will  it  cost  to  build  a  cement  walk  6  ft. 
wide  around  a  block  500  ft.  square  at  lOi  ct.  per  square 
foot  ? 

39.  If  a  tight  board  fence  6  ft.  high  is  built  around  the 
same  block  2  ft.  inside  of  the  walk,  how  will  its  area  com- 
pare with  that  of  the  walk  ? 

40.  What  must  be  the  depth  of  a  cistern  6  ft.  in  diame- 
ter which  shall  contain  600  gal.,  if  a  gallon  of  water  weighs 
10  lb.  and  a  cubic  foot  of  water  weighs  1000  oz.  ? 


238  EXERCISES  FOR  REVIEW 

41.  If  the  pressure  of  the  atmosphere  at  the  surface  of 
the  earth,  when  the  barometer  stands  at  30  in.,  is  about 
15  lb.  to  the  square  inch,  Avhat  is  the  pressure  on  the 
human  body  if  its  surface  is  16  sq.  ft.  ?  What  would  be 
the  difference  in  pressure  if  the  barometer  stood  at  29  in.  ? 

42.  How  many  grains  of  gold  are  there  in  6  lb.  4  oz. 
5  pwt.  ? 

43.  If  employed  6  da.  in  the  week  and  8  hr.  daily,  how 
long  would  it  take  to  count  1 50000000  at  the  rate  of  |100 
a  minute  ? 

44.  If  sound  travels  at  the  rate  of  1100  ft.  per  second, 
and  the  report  of  a  gun  is  heard  10  sec.  after  the  appear- 
ance of  the  smoke,  how  far  distant  is  the  observer  ? 

45.  What  number  between  300  and  400  is  exactly  divis- 
ible by  2,  3,  4,  5  ? 

46.  If  4  cu.  in.  of  iron  weigh  a  pound,  find  the  weight 
of  a  rectangular  vessel  an  inch  and  a  half  thick  without  a 
top,  the  vessel  being  lOJ  ft.  by  8^  ft.  by  5J  ft.  outside 
measure. 

47.  A  cubic  foot  of  copper  weighs  556 J  lb.,  and  can  be 
drawn  into  a  wire  1  mi.  125  rd.  long.  Find  the  weight  of 
copper  necessary  for  a  wire  60  mi.  long  and  also  the  area 
of  a  cross  section  of  the  wire. 

48.  How  long  is  an  iron  bar  containing  a  cubic  foot  of 
iron  if  its  dimensions  are  |  of  an  inch  by  ~|  of  an  inch  ? 

49.  If  a  cubic  foot  of  water  weighs  1000  oz.,  find  the 
number  of  grains  in  a  cubic  inch„ 

50.  Explain  whether  0.023  or  0.024  is  more  nearly  equal 
to  0.02349  and  state  in  words  the  error  in  excess  or  defect 
in  each  case. 


LWhiKisKs  Foil  ni:vii:\v  2oli 

51.  Divide  0.84827  by  0.23  correct  to  0.01. 

52.  Multiply  3.1-tol)  by  1G.325  correct  to  n.l. 

53.  If  the. meter  is  39.3708  in.,  what  part  of  a  meter  is 
a  yard  ? 

54.  If  the  average  length  of  a  degree  of  latitude  is 
3G5000  ft.,  tind  the  length  of  a  meter  in  feet  and  inches. 

55.  If  water  expands  10 ^fv  when  it  freezes,  how  much 
does 'ice  contract  when  it  turns  into  water? 

56.  Find  the  discount  of  -^1000  for  90  da.  at  Qf.  Show 
that  the  interest  on  this  discount  for  the  same  time  is 
equal  to  the  difference  between  the  interest  and  the  dis- 
count of  81000. 

57.  ShoAV  that  the  interest  on  the  discount  of  $1000  for 
one  year  at  6^  is  the  same  as  the  discount  on  the  interest 
at  the  same  rate  for  the  same  time. 

58.  If  a  person  saves  8300  a  year,  and  invests  his  sav- 
ings at  4 5^  compound  interest  for  10  yr.,  what  amount 
does  he  accumulate  ? 

59.  Which  is  the  better  investment,  bonds  bousfht  at 
112  yielding  %Jo  interest,  or  stocks  bought  at  85  yielding 
4^  dividends  ? 

60.  A  person  owns  302  810  shares  of  Wolverine  Port- 
land Cement  Stock,  paying  a  semiannual  dividend  of  5^; 
20  shares  of  bank  stock  of  8100  each,  paying  a  semiannual 
dividend  of  2|/o;  30  Mexican  Plantation  Bonds  of  8300 
each,  paying  7/o  interest.  What  is  his  total  annual  income 
from  these  sources  ? 

61.  A  merchant  adds  33 J ^o  to  the  cost  price  of  his  goods, 
and  gives  his  customers  a  discount  of  10  f. :  what  profit  does 
he  make? 


240  EXERCISES  FOli   HE  VIEW 

62.  If  a  ship  sails  from  San  Francisco  Oct.  15  and 
reaches  Japan  after  20  da.,  what  is  the  date  of  her 
arrival  ? 

63.  When  it  is  2  p.m.  Sunday,  Feb.  15,  at  Greenwich, 
what  time  and  date  is  it  at  longitude  165°  W.  ? 

64.  The  Canadian  Pacific  Railway  uses  twenty-four-hour 
clocks  (hours  from  noon  to  midnight  are  12  to  2-4  o'clock) 
at  Port  Arthur  and  west.  When  it  is  20  o'clock,  standard 
time,  at  Winnipeg,  what  time  is  it  at  Toronto  ? 

65.  A  street  40  ft.  wide  is  to  be  paved  for  a  distance 
of  1680  ft.  If  it  costs  32  ct.  a  cubic  yard  for  excavating 
to  a  depth  of  2  ft.,  4  ct.  a  square  yard  for  sand  cushion, 
11.17  a  square  yard  for  crushed  stone  filling,  and  48|  ct. 
a  square  yard  for  concrete,  what  is  the  cost  of  the  paving  ? 

66.  Dec.  28,  1886,  Mr.  Harvey  insured  his  life  for 
13000  on  the  fifteen-payment  life  plan,  paying  a  quarterly 
(z.e.  four  times  a  year)  premium  of  !^44.10.  Instead  of 
continuing  the  insurance  at  the  end  of  the  15  yr.,  he  accepts 
a  cash  settlement  of  12942.20.  Allowing  $15  a  year  per 
11000  for  protection  afforded,  what  rate  of  interest  has 
his  money  earned  ? 

67.  In  1903  Michigan  levied  a  tax  of  $397525  for  the 
support  of  the  State  University  at  the  rate  of  ^  of  a  mill. 
What  was  the  valuation  of  the  state  property  ? 

68.  How  many  tons  of  coal  will  a  bin  10  ft.  by  6|  ft. 
by  7|  ft.  hold  if  one  ton  occupies  36  cu.  ft.  ? 

69.  Simplify  2  of  I  ^  2f  -f-  5i  X  ^'V- 

70.  Find  the  least  fraction  that  added  to  |,  -^^  and  ^g 
will  make  the  result  an  integer. 

„,     c,.      ,.f     4.561      0.0075 

71.  Simplify  ^^  X  ^j^. 


EXERCISES  FOR  REVIEW  241 

72.  A  person's  income  is  -li^  2500  a  year.  He  spends  on 
an  average  '"5^27.75  a  week.  If  lie  deposits  las  savings  in 
a  bank  ev-ery  3  mo.,  how  mucli  will  he  accumulate  in  10  yr. 
if  the  bank  pays  3%  compound  interest? 

73.  How  many  miles  are  there  in  10000  ft.  and 
1000000  in.  ? 

74.  Use  short  methods  in  finding  the  product  of  14  x  70, 

309  X  81,  4728  x  998,  85  x  85,  67  x  73. 

75.  Find  by  factors  the  square  root  of  44100,  1352,  225. 
Find  the  square  root  of  these  numbers  by  logarithms. 

76.  The  distance  between  two  places  on  a  map  is 
207""".  What  is  the  distance  in  kilometers  if  the  scale  of 
the  map  is  1  to  10000  ? 

77.  A  copper  wire  2  yd.  1.23  ft.  long  is  cut  into  pieces 
0.022  of  a  foot  long.  How  many  pieces  will  there  be,  and 
what  length  will  be  left  over  ? 

78.  How  many  rolls  of  paper  20  in.  wide  and  12  yd. 
long  will  be  required  to  paper  a  room  10  ft.  long,  12  ft. 
wide,  and  9  ft.  high,  allowing  96  sq.  ft.  for  windows  and 
doors  ? 

79.  Find  the  specific  gravity  of  a  substance  that  weighs 
12^  in  air  and  7^  in  water. 

80.  A  pound  Troy  is  what  per.  cent  of  a  pound  avoir- 
dupois ? 

81.  What  are  the  proceeds  of  a  note  for  '*?1250  at  5%, 
dated  Oct.  17, 1905,  at  3  mo.,  and  discounted  Dec.  1  at  6%? 

82.  If  a  liter  of  air  weighs  1.29^,  find  the  weight  of 
air  in  a  room  40  ft.  by  30  ft.  by  12  ft. 

LYMAX'S  ADV.  AR.  —  16 


'24:2  EXERCISES  FOR  REVIEW 

83.  If  sound  travels  at  the  rate  of  1090  ft.  per  second, 
how  far  distant  is  a  thundercloud  when  the  sound  of  the 
thunder  follows  the  flash  of  lightning  after  6  sec.  ? 

84.  A  merchant  sold  some  goods  and  took  in  payment 
a  90-da.  note  at  5%,  dated  July  10,  1905.  Aug.  5  he  dis- 
counted the  note  at  the  bank  at  6%.  What  were  the  pro- 
ceeds of  the  note  ? 

85.  Twenty-five  loads  of  gravel  are  spread  uniformly 
over  a  path  200  ft.  long  and  5  ft.  wide.  What  is  the 
depth  of  the  gravel,  a  load  being  1  cu.  yd.  ? 

86.  If  a  half  of  a  liter  of  a  given  substance  weighs 
1500^,  what  is  the  specific  gravity  of  the   substance  ? 

87.  Find  the  exact  interest  on  ^500  from  July  3  to 
Sept.  10  at  6%. 

88.  A  wholesale  dealer  sold  goods  at  a  discount  of  25%, 
10%  and  3%  for  cash.  He  received  in  payment  -^3269.75. 
What  was  the  list  price  of  the  goods  ? 

89.  When  U.  S.  3's  can  be  bought  at  108  (brokerage  \\ 
how  many  bonds  can  be  bought  for  $4325  ? 

90.  The  nearest  fixed  star  is  estimated  to  be  23000000- 
000000  mi.  distant.  How  many  years  does  it  take  light  to 
travel  this  distance  at  the  rate  of  186000  mi.  a  second  ? 

91.  On  a  note  for  15000,  dated  Jan.  4,  1904,  due  in  1  yr. 
with  interest  at  6%,  payments  of  f  100  had  been  made  on 
the  4th  of  each  month  for  11  mo.  in  succession.  What 
amount  was  due  Jan.  4,  1905  ? 

92.  What  must  be  the  face  of  a  note  at  90  da.  so  that 
the  borrower  shall  receive  $1000,  the  discount  being  at 
the  rate  of  7%  per  annum  ? 


EXERCISES  FOR   REVIEIV  243 

93.  A  note  for  81000,  due  in  1  yr.  at  5%,  has  an  indorse- 
ment of  ^'2d0  made  5  mo.  after  date.  W'Jiat  is  the  amount 
due  at  the  end  of  the  year  ? 

94.  A  note  for  1 500,  dated  March  1,  1903,  and  payable 
2  yr.  from  date,  with  interest  at  G%  per  annum,  has  on  it 
the  following  indorsement's:  April  1,  1903,  850;  June  1, 

1903,  850;    Sept.  1,  1903,  820;    and  May  1,  1904,  850. 
What  amount  is  due  March  1,  1905? 

95.  A  note  for  82000,  dated  May  15,  1903,  at  5%  per 
annum,  has  the  following  indorsements:  July  1,  1903, 
860;  Aug.  1,  1903,810;  Oct.  1,  1903,  820;  Jan.  2,  1904, 
8100;    May  15,  1904,  8100;    Sept.  1,  1904,  820;    Nov.  1, 

1904,  820;   May  15,  1905,  8200.      What  amount  is  due 
Jan.  2,  1906  ? 

96.  If  bank  stock  pays  a  7%  annual  dividend,  at  what 
price  must  it  be  bought  to  yield  a  5%  income  on  the 
investment  ? 

97.  A  traveler  bought  in  New  York  a  bill  of  exchange 
on  London  for  £500,  exchange  being  at  4.87.  How  much 
did  he  pay  the  banker  ? 

98.  The  number  of  thousands  of  people  who  emigrated 
annually  from  Ireland  between  and  including  1876  and 
1885  Avere  as  follows:  37.5,  38.5,  41.1,  47,  95.5,  78.4, 
89.1,  108.7,  75.8,  62.     Illustrate  graphically. 

99.  The  annual  premiums  charged  by  one  of  the  leading 
life  insurance  companies  at  certain  ages  to  insure  the  pay- 
ment of  81000  at  death  are  as  follows: 

Age  21  24  27  30  35  40  45  50 

Premium  1 19.53  $20.86  $22.40  $24.18  $27.88  $32.76  $39.36  $48.39 

Illustrate  grapliically  and  determine  the  probable  pre- 
miums at  ages  25,  33,  and  48. 


244  EXERCISES  FOR  REVIEW 

NEW  YORK   STATE   REGENTS'   EXAMINATIONS 

The  following  exercises  are  taken  from  the  Regents' 
examination  questions  in  advanced  arithmetic  for  the 
state  of  New  York : 

1.  Columbus  discovered  America  Oct.  12,  1492.  Ex- 
plain why  we  celebrated  the  400th  anniversary  Oct.  23, 
1892. 

2.  Find  the  prime  factors  of  each  of  the  following 
numbers :  42,  48,  126,  144.  Indicate  the  combination  of 
factors  necessary  to  produce  (a)  the  greatest  common 
divisor  of  these  numbers,  (&)  their  least  common  multiple. 

3.  Find  the  number  of  square  yards  in  the  four  walls 
and  ceiling  of  a  room  16|-  ft.  long,  13|^  ft.  wide,  and  9  ft. 
high,  making  no  allowance  for  openings. 

4.  Make  a  receipted  bill  of  the  following:  William 
Stone  buys  this  da}^,  of  Flagg  Brothers,  2  bbl.  of  flour  at 
15.50,  20  lb.  sugar  at  5^  ct.,  4  lb.  coffee  at  35  ct.,  5  lb. 
butter  at  28  ct.,  2  bu.  potatoes  at  45  ct. 

5.  Simplify    ^^T5x  0.5 +  325 -0.331x21  ^^^^^^  ^ 

^     ^  0.25+0.049+0.014  ^ 

the  result  both  as  a  common   fraction  and  as  a  decimal 
fraction. 

6.  A  man  walks  8|  mi.  in  2  hr.  20  min.  How  long 
will  it  take  him  to  walk  111  mi.  ?  (Solve  botli  by  analysis 
and  by  proportion^) 

7.  How  many  liters  of  water  will  be  contained  in  a 
vessel  whose  base  is  1"'  square  and  whose  depth  is  6'^'"  ? 

8.  A  merchant  sold  goods  for  11125;  half  he  sold  at 
an  advance  of  25%  on  the  cost,  two  fifths  at  an  advance 
of  121  %  j^i-,(^[  the  remainder  at  \  the  cost.  How  much  did 
he  pay  for  the  goods? 


NE]V   YOliK  STATE  UEGENTH'  EXAMINATIONS      245 

9.    Two  successive  discounts  of  15%  and  10  %  reduced 
a  bill  to  $489.60.     What  was  the  orio-inal  bill  ? 

10.  Find  the  proceeds  of  a  note  for  'If  500,  payable  in 
90  da.,  with  interest  at  6%,  if  discounted  at  a  bank  at 
6%,  40  da.  after  date. 

11.  A  house  and  lot  cost  f  5000;  the  insurance  is  ^25, 
taxes  are  $50  and  repairs  #75  annually.  What  rent  must 
be  received  in  order  to  realize  6%  on  the  investment? 

12.  At  what  price  must  5%  bonds  be  bought  so  as  to 
realize  7|^%  on  the  investment? 

13.  Find  the  square  root  of  243.121  correct  to  three 
decimal  places. 

14.  Three  families,  consisting  of  3,  4,  and  5  persons  re- 
spectively, camped  out  during  the  summer  months,  agree- 
ing that  the  expenses  should  be  divided  in  the  ratio  of 
the  number  of  persons  in  each  family.  The  expenses 
amounted  to  1606.  What  number  of  dollars  should  each 
family  pay? 

15.  The  diagonal  of  a  square  field  is  40  rd.  How  many 
acres  does  the  field  contain  ? 

16.  A  schoolhouse  costing  $9500  is  to  be  built  in  a 
district  whose  property  is  valued  at  $1920000.  Find 
(a)  the  rate  of  taxation,  (^)  the  amount  of  tax  to  be  paid 
by  a  man  Avhose  property  is  valued  at  $6500. 

17.  A  sight  draft  on  New  York  was  sold  in  St.  Louis 
for  $3542,  exchange  being  |%  premium.  liequired  the 
face  of  the  draft. 

18.  Which  would  be  the  better,  to  invest  $4356.25  in 
industrial  4's  at  87,  brokerage  |^,  or,  with  the  same  sum, 
to  purchase  real  estate  which  jdelds  an  annual  rental  of 
$300? 


246  EXERCISES  FOR   REVIE]V 

19.  On  a  note  for  1700,  dated  Oct.  15,  1898,  due  in 
one  year,  with  interest  at  5%,  tlie  following  payments  have 
been  made:  March  9,  1899,^300;  June  1,  1899,  |250. 
Find  the  amount  due  at  maturity. 

20.  A  house  worth  $12000  was  insured  for  J  of  its 
value  by  three  companies ;  the  first  took  J  of  the  risk  at 
1%,  tiie  second  ^  of  the  risk  at  |%,  and  the  third  the 
remainder  at  |%.      What  was  the  whole  premium  paid? 

21.  Find  the  trade  discount  on  a  bill  of  goods  amount- 
ing at  list  price  to  1360,  but  sold  30%,  8%  and  5%  off.    • 

22.  (a)  22-1-  is  what  per  cent  of  71?  (5)  What  per 
cent  of  5  lb.  avoirdupois  is  7|-  oz.  ?  (<?)  -f^  is  225%  of 
what  number? 

23.  The  specific  gravity  of  copper  is  8.9,  of  silver  10.5, 
and  in  an  alloy  of  these  metals  the  weight  of  the  copper  is 
to  the  weight  of  the  silver  as  5:6.  Find, the  ratio  of 'the 
bulk  of  copper  in  the  alloy  to  that  of  the  silver. 

24.  How  many  kilograms  of  water  are  required  to  fill  a 
tank  2"'  deep  whose  base  is  a  regular  hexagon  0.4"'  on  a 
side  ? 

25.  A  horse  costs  three  times  as  much  as  a  buggy,  and 
the  harness  and  robes  cost  one  half  as  much  as  the  horse. 
If  the  total  cost  was  $330,  what  was  the  cost  of  each? 
Write  an  analysis. 

26.  Reduce  the  couplet  9| ;  32^2  to  the  integral  form 
in  lowest  terms. 

27.  What  is  the  heiglit  of  a  wall  which  is  14J  yd.  in 
length  and  -^^  of  a  yard  in  thickness,  and  which  cost  -it^lOC), 
it  having  been  paid  for  at  the  rate  of  $10  per  cubic  yard? 

28.  Find  the  cost,  at  $15  per  M,  of  75  pieces  of  lumber 
each  14  ft.  by  16  in.  by  If  in. 


NEW    YORK  STATE  REfUCNTS     EXAMLWATIONS      247 

29.  Find  tlie  prime  factors  of  18902. 

30.  TIkj  diameters  of  tlie  wliecls  of  tliree  bicycles  are 
24  in.,  82  in.  and  34  in.  respectively.  Each  luis  a  ribljoii 
tied  to  the  fop  of  the  wheel.  J  low  far  must  the  l)i(;ycles 
go  that  the  ribbons  may  be  again  in  the  same  relative 
positions? 

31.  If  a  boy  buys  peaches  at  the  rate  of  5  for  2  ct.,  and 
sells  them  at  the  rate  of  4  for  3  ct.,  how  many  must  he  buy 
and  sell  to  make  a  profit  of  '$4.20  ? 

32.  Give  a  method  of  (a)  proving  addition  ;  (^)  sub- 
traction ;   (c)  multiplication ;   (ri)  division. 

33.  Express  by  signs  of  per  cent,  by  a  decimal,  and 
by  a  common  fraction  in  its  lowest  terms,  each  of  the 
following:  (a)  -^\  per  cent;  (Z>)  4|  %  ;  (/?)  five  sixty- 
fourths  ;  (cZ)  three  thousand  one  hujidred  jB.fteen  thou- 
sandths. 

34.  Write  a  number  that  shall  be  at  the  same  time 
simple,  composite,  abstract  and  even.  State  why  it  fills 
each  of  these  requirements, 

35.  Add  together  15262986957  and  3879,  and  multiply 
the  19th  part  of  the  sum  by  76. 

36.  In  trying  numbers  for  factors,  why  is  it  unnecessary 
to  try  one  larger  than  the  square  root  of  the  number? 

37.  Find  the  cost,  at  25  ct.  a  rod,  of  building  a  fence 
round  a  square  10-acre  field. 

38.  How  many  cords  of  wood  can  be  stored  in  a  shed 
16  ft.  long,  12  ft.  wide  and  6  ft.  high? 

39.  Find  the  sum  of  11,  |  x  1-|,  3,  -^q.  Express  the 
result  as  a  decimal. 


248  EXERCISES  FOR  REVIEW 

40.  If  I  sell  I  of  a  larm  for  what  -|  of  it  cost,  what  is  my 
per  cent  of  gain  ? 

41.  I  sell  goods  at  15%  below  the  market  price  and  still 
make  a  profit  of  10%.  What  per  cent  above  cost  was  the 
market  price  ? 

42.  How  was  the  principal  unit  of  the  metric  S5^stem 
determined  ?  Explain  the  relation  between  this  unit  and 
the  metric  units  of  capacity  and  weights. 

43.  Find  the  cube  root  of  4.080659192. 

44.  Prove  that  the  product  of  any  three  consecutive 
numbers  is  divisible  by  6  or  by  24.  Determine  when  it  is 
divisible  by  6  ;   when  it  is  divisible  by  24. 

45.  The  diameters  of  four  spheres  are  3.75,  5,  6.25  and 
7.5.  Prove  that  the  volume  of  one  of  them  is  equal  to  the 
volume  of  the  remaining  three. 

46.  A  merchant  buys  goods  to  the  amount  of  $4000  ;  in 
order  to  pay  for  them  he  gets  his  note  for  60  da.  dis- 
counted at  a  bank.  If  the  face  of  the  note  is  $4033.61, 
what  is  the  rate  of  discount  ? 

47.  Prove  that  the  exact  interest  of  any  sum  for  a  given 
number  of  days  is  equal  to  the  interest  of  the  same  sum  for 
the  same  number  of  days  (as  usually  computed)  diminished 
by  y^g-  of  itself. 

48.  A  sells  a  certain  amount  of  5%  stock  at  86  and 
invests  in  6%  stock  at  103;  by  so  doing  his  income  is 
changed  $1.  What  amount  of  stock  did  he  sell?  Was 
his  income  increased  or  diminished  ? 

49.  Divide  |  by  |-  and  demonstrate  the  correctness  of 
tlie  work. 


NEW  YOBK  STATE  UEGENTS'   EXAMINATIONS      249 

50.  Multiply  42.35  by  3.14159,  using  tlie  contracted 
method  and  finding  the  result  correct  to  two  decimal 
places.  Prove  the  work  by  division,  using  the  contracted 
method. 

51.  A  man  borrows  >^4500,  and  agrees  to  pay  princi[)al 
and  interest  in  four  equal  annual  installments.  If  the 
rate  of  interest  is  6%,  what  will  be  the  amount  of  each 
annual  payment  ? 

52.  AVhen  it  is  Monday,  7  A.M.,  at  San  Francisco,  longi- 
tude 122°  24'  15"  W.,  what  day  and  time  of  day  is  it  at 
Berlin,  longitude  13°  23'  So"  E.  ? 

53.  When  exchange  is  at  5.18,  find  the  gain  on  100"^ 
of  silk  bought  in  Paris  at  2  francs  a  meter  and  sold  in 
New  York  at  89  ct.  a  3^ard,  the  duty  being  6%  ad  valorem. 

54.  Find  the  face  of  a  sisfht  draft  that  can  be  bou"-ht 

o  o 

for  '1)585.80  when  exchange  is  at  a  premium  of  |%. 

55.  Divide  0.8487432  by  0.075637  and  multiply  the 
quotient  by  0.835642.  Find  the  result  correct  to  three 
decimal  places,  using  the  contracted  methods  of  division 
and  multiplication  of  decimals. 

56.  Express  in  Avords  each  of  the  following:  600.035, 
u.uo^,  ouogQQQ,  5000,  looo- 

57.  A  body  on  the  surface  of  the  earth  weighs  27  lb. 
Assuming  that  the  radius  of  the  earth  is  4000  mi.,  find  the 
weight  of  the  same  body  2000  mi.  above  the  surface.  (The 
weight  of  a  body  above  the  surface  of  the  earth  varies  in- 
versely as  the  square  of  the  distance  from  the  center  of  the 
earth.) 

58.  Washington  is  77°  3'  W.  longitude  and  Pekin  116° 
29'  E.  longitude.  When  it  is  9.30  p.m.,  Tuesday,  Dec.  31, 
1901,  at  Washington,  what  is  the  time  of  the  day,  the  day 
of  the  week,  and  the  date  at  Pekin  ? 


250  EXERCISES  FOR  REVIEW 

59.  Find  the  exact  interest  on  1590  from  Sept.  18, 1893, 
to  March  1,  1894,  at  4|%. 

60.  Is  the  merchants'  rule  or  tlie  United  States  rule  for 
computing  partial  payments  more  favorable  to  the  debtor  ? 
Give  reasons. 

61.  A  locomotive  runs  |  of  a  mile  in  |  of  a  minute.  At 
what  rate  au  hour  does  it  run  ?     (Give  analysis  in  full.) 

62.  The  edges  of  a  rectangular  parallelopiped  are  in  the 
proportion  of  3,  4  and  6;  its  volume  is  720  cu.  in.  Find 
its  entire  surface. 

63.  A  note  for  8250,  due  in  1  yr.,  with  interest  at  6%, 
is  dated  Jan.  1,  1892.  What  is  the  true  value  of  this  note 
Oct.  1,  1892  ? 

64.  At  10  A.M.  Jan.  5  a  watch  is  5  min.  too  slow;  at 
2  P.M.  of  Jan.  9  it  is  3  min.  20  sec.  too  fast.  When  did 
it  mark  correct  time  ? 

65.  A  gallon  contains  231  cu.  in. ;  a  cubic  foot  of  water 
weighs  62.5  lb.;  mercury  is  13.5  times  as  heavy  as  Avater. 
How  many  gallons  of  mercury  will  weigh  a  ton  ? 

66.  Find  the  face  of  a  note  that  will  yield  1 861.44  pro- 
ceeds when  discounted  for  90  da.  at  6%. 

67.  A  merchant  buys  goods  listed  at  12500,  getting 
successive  trade  discounts  of  20,  10  and  5 ;  he  sells  his 
goods  at  20%  above  the  cost  price,  taking  in  payment  a 
note  at  60  da.  without  interest ;  he  then  gets  the  note 
discounted  at  6%  and  pays  his  bill.     Find  his  entire  gain. 

68.  A  person  deposits  $100  a  year  in  a  savings  bank 
that  pays  4%  interest,  compounded  annually.  How  much 
money  stands  to  his  credit  immediately  after  the  fifth 
deposit  ? 


NEW   YORK   STATE   llEGENTS'   EXAMINATIOXS      -2.^)1 

69.  Cluin«;t'  '2()^)'j-]'2  ill  tliu  (|uiiiai'y  scale  to  an  e<[uivalent 
number  in  the  deeinial  scale,  and  prove  the  work. 

70.  A  New  York  merchant  remitted  to  London  throui^di 
his  broker  £12000  18s.  [hi  Find  the  cost  of  the  draft  if 
exchange  is  at  4.81>|  and  brokerage  is  {%. 

71.  In  extracting  the  cube  root  state  and  explain  the 
process  of  («)  separating  into  periods,  (b)  forming  the 
trial  divisor,  (^)  completing  the  divisor. 

72.  A  merchant  buys  goods  at  a  list  price  of  8800, 
o-ettincT  discounts  of  10,  20  and  5  with  60  da.  credit, 
or  a  further  discount  of  5%  for  cash.  How  much  will  he 
gain  by  borrowing  at  ()%  to  pay  the  bill? 

73.  At  a  certain  election  510  votes  were  cast  for  two 
candidates;  |  of  those  cast  for  one  equaled  |  of  those 
cast  for  the  other.  How  many  votes  were  cast  for  each 
candidate  ? 

74.  If  the  cost  of  an  article  had  been  S%  less,  the  gain 
would  have  been  10%  more.     What  was  the  per  cent  gain  ? 

75.  Prove  that  the  excess  of  9's  in  the  product  of  two 
numbers  is  equal  to  the  excess  in  the  product  of  the 
excesses  in  the  two  factors. 

76.  Derive  a  rule  for  marking  goods  so  that  a  given 
reduction  may  be   made    from    the    marked  price    and   a 

given  profit  still  made  on  the  cost. 

77.  The  greatest  common  divisor  and  the  least  common 
multiple  of  two  numbers  between  100  and  200  are  respec- 
tively 6  and  3150.     Find  the  numbers. 

78.  How  much  will  the  product  of  two  numbers  be  in- 
creased by  increasing  each  of  the  numbers  by  1  ?  Give 
proof. 


252  EXERCISES  FOR  REVIEW 

79.  The  longer  sides  of  an  oblong  rectangle  are  15  ft. 
and  the  diagonal  is  20  ft»     Find  its  area. 

80.  Find  the  fourth  term  of  the  following  proportion 
and  demonstrate  the  principle  on  which  the  operation  is 
based  :  8  :  12  =  10  :  rr. 

81.  Demonstrate  the  following:  If  the  greater  of  two 
numbers  is  divided  by  the  less,  and  the  less  is  divided  by 
the  remainder,  and  this  process  is  continued  till  there  is 
no  remainder,  the  last  divisor  will  be  the  greatest  common 
divisor. 

82.  Find  in  inches  to  two  places  of  decimals  the  diagonal 
of  a  cube  whose  volume  is  9  cu,  ft. 

83.  Compare  the  standard  units  of  money  of  the  United 
States,  England,  France,  and  Germany  as  to  relative  value. 
Find  the  value  of  $100  in  each  of  the  other  units. 

84.  A  dealer  sent  a  margin  of  81500  to  his  broker, 
April  16,  1905,  and  ordered  him  to  buy  100  shares  of 
American  Sugar  stock.  The  broker  filled  the  order  at 
131 1  and  sold  the  stock  May  1  at  1261  charging  1% 
brokerage  each  way  and  6%  interest.  How  much  money 
should  be  returned  to  the  dealer  ? 

85.  A  four  months'  note  for  1 584,  without  interest,  is 
discounted  at  a  bank  at  5%  on  the  day  of  its  issue.  Find 
the  proceeds  of  the  note. 

86.  What  is  the  difference  between  a  discount  of  10% 
and  two  successive  discounts  of  5%  each  on  a  bill  of  $832  ? 

87.  If  I  buy  cloth  at  $1.20  a  yard,  how  must  I  sell  it  so 
as  to  gain  25%  ? 

88.  Find  the  cost  of  paving  a  walk  140"^™  wide  and  |  of 
a  kilometer  long  at  $1.25  a  square  meter. 


NEW   YORK   STATE  liEGENTS'   EXAMINATIONS      253 

89.  Indicate  tlie  factors  which,  multiplied  together, 
equal  the  square  root  of  441. 

90.  A  newsboy  buys  144  daily  papers  at  20  ct.  a  dozen, 
and  sells  them  at  8  ct.  each.  At  the  end  of  0  da.  lie  has 
18  papers  on  hand.  How  much  has  he  made  (lurin<^r  the 
week  ? 

91.  The  diameters  of  two  concentric  circles  are  20  ft. 
and  30  ft.      Find  the  area  of  the  ring. 

92.  What  yearly  income  will  S^  2267.50  produce  when 
invested  in  U.  S.  4's  at  113|,  brokerage  J%  ? 

93.  Find  the  amount  of  8486.50  for  1  yr.  5  mo.  and 
17  da.  at  5|%  simple  interest. 

94.  I  buy  stocks  at  4%  discount  and  sell  at  4%  pre- 
mium ;  what  per  cent  proht  do  I  make  on  the  investment  ? 

95.  A  merchant  buys  goods  to  the  amount  of  f  1575  on 
9  months'  credit;  he  sells  them  for  81800  ca^h.  Money 
being  Avortli  6%,  how  much  does  he  gain? 

96.  Find  the  cost,  at  60  ct.  a  yard,  of  carpeting  a  room 
16  ft.  4  in.  wide  and  21  ft.  6  in.  long  with  carpet  27  in. 
wide,  if  the  strips  of  carpet  run  lengthwise. 

97.  Find  the  cost  at  45  ct.  a  roll  of  papering  the  walls 
of  a  room  16|-  ft.  long,  15  ft.  wide,  and  12  ft.  high,  mak- 
ing no  allowance  for  openings. 

98.  Find  the  cost  of  plastering  the  four  walls  and  the 
ceiling  of  a  room  15  ft.  long,  12  ft.  wide  and  9  ft.  high  at 
15  ct.  a  sq.  yd.,  allowing  6  sq.  yd.  for  openings. 


Approved  Text-Books  in  Algebra 

By  WILLIAM  J.  MILNE,  Ph.D..  LL.D. 

President  of  New  York  State  Normal  College 


MILNE'S  GRAMMAR  SCHOOL  ALGEBRA     50  cents 

This  work,  desit^nctl  especially  for  j^rainmar  schools,  is  adapted  for  the 
use  of  beginners  in  either  public  or  private  schools.  It  is  somewhat  easier 
and  less  advanced  than  the  Elements  of  Algebra  by  the  same  author. 
The  fundamental  principles  of  the  science  of  algebra  are  presented  in 
such  a  manner  that  a  deep  interest  in  the  study  is  awakened  at  once. 

MILNE'S  ELEMENTS  OF  ALGEBRA     .    .     60  cents 

This  book  is  intended  to  lay  a  sound  foundation  for  more  advanced  wcjrk 
in  the  study.  Some  of  the  distinctive  features  are  : — The  easy  and  natu- 
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and  judicious  selection  of  examples  for  practice;  and  the  early  introduction 
and  practical  use  of  the  equation,  which  is  made  the  keynote  of  the  book. 

MILNE'S  HIGH  SCHOOL  ALGEBRA.    .    .    .     $1.00 

This  text-book  provides  a  complete  course  for  high  schools  and  acad- 
emies. It  covers  fully  all  the  subjects  required  for  entrance  by  any 
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and  progressive  steps  to  a  clear  comprehension  of  the  principles  of  the 
science,  and  then  receives  a  thorough  drill  in  applying  these  principles. 

MILNE'S  ACADEMIC  ALGEBRA $1.25 

In  this  book,  the  treatment  of  the  subject  throughout  is  based  upon 
the  most  modern  presentation  of  the  science.  It  meets  fully  the  most 
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their  truth  by  the  method  of  deductive  reasoning. 

MILNE'S  ADVANCED  ALGEBRA      ....     $1.50 

This  book  represents  the  most  modern  presentation  of  the  science,  and 
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AMERICAN    BOOK    COMPANY 

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A   Complete  System   of  Pedagogy 

IN    THREE    VOLUMES 
By  EMERSON    E.  WHITE,  A.M.,  LL.D. 


THE  ART  OF  TEACHING.     Cloth,  321   pages      .         .       Price,  $1.00 

This  new  work  in  Pedagogy  is  a  scientific  and  practical  considera- 
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book  study,  class  instruction  and  management,  examinations,  promotion 
of  pupils,  etc. 

ELEMENTS  OF  PEDAGOGY.     Cloth,  336  pages     .         .     Price,  $1.00 

This  treatise,  by  unanimous  verdict  of  the  teachers'  profession,  has 
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SCHOOL    MANAGEMENT.     Cloth,  320  pages      .         .       Price,  $1.00 

The  first  part  of  this  work  is  devoted  to  school  organization  and 
discipline,  and  the  second  part  to  moral  training.  Principles  are  clearly 
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own  wide  experience.  A  clear  light  is  thrown  on  the  most  important 
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all  teachers  who  feel  the  importance  of  this  work. 


Copies  sent,  prepaid,  to  any  address  o>i  receipt  of  the  price. 

American   Book   Company 

New  York  ♦  Cincinnati  ♦  Chicago 

(200) 


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